If you find this useful or would be interested in providing feedback and suggestions, please let me know at dwharder@gmail.com. See my other related pages on nice eigenvalue decompositions (eigendecompositions), integer matrices with integer eigenvalues, integer matrices with integer singular values, integer-normed vectors, integer-normed complex vectors, geometric sequences with nice norms.

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When looking for examples in linear algebra courses, it is often preferable to have examples that have nice representations on the blackboard. Consequently, if the norm contains a surd, it becomes difficult to work with, and these examples are problematic in the class room. Here are integer vectors that have integer norms. In some cases, it is possible to have orthonormal bases using ratios of simple vectors; these, too, are presented.

• Two-dimensional vectors
• Generating mutually-orthogonal two-dimensional vectors
• Normalized two-dimensional vectors with terminating decimal representation
• Other two-dimensional integer vectors with integer norms
• Three-dimensional vectors
• Generating mutually-orthogonal three-dimensional vectors
• Normalized three-dimensional vectors with terminating decimal representation
• Mutually orthogonal normalized three-dimensional vectors with terminating decimal representation
• Other three-dimensional integer vectors with integer norms
• Four-dimensional vectors
• Generating mutually-orthogonal four-dimensional vectors
• Other mutually-orthogonal four-dimensional vectors
• Normalized four-dimensional vectors with terminating decimal representation
• Other four-dimensional integer vectors with integer norms
• Five-dimensional vectors
• Normalized five-dimensional vectors with terminating decimal representation
• Other five-dimensional integer vectors with integer norms
• Mutually-orthogonal five-dimensional vectors
• Six-dimensional vectors
• Normalized six-dimensional vectors with terminating decimal representation
• Other six-dimensional integer vectors with integer norms

Two-dimensional vectors

Generating mutually-orthogonal two-dimensional vectors

Given any one vector $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ in ${\bf R}^2$, it is always possible to construct two mutually orthogonal vectors as follows:

$\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ and either $\begin{pmatrix} \beta \\ -\alpha \end{pmatrix}$ or $\begin{pmatrix} -\beta \\ \alpha \end{pmatrix}$

Therefore, if the first vector is normalized, so is the second. If normalized, an orthogonal matrix therefore, is

$\begin{pmatrix} \alpha & \beta\\ \beta & -\alpha \end{pmatrix}$ or $\begin{pmatrix} \alpha & \beta\\ -\beta & \alpha \end{pmatrix}$

Normalized two-dimensional vectors with terminating decimal representation

Vector	Normalized vector	Norm
[  3    4]	[0.6     0.8]	5
[  7   24]	[0.28    0.96]	25
[ 44  117]	[0.352   0.936]	125
[336  527]	[0.5376  0.8432]	625
[237 3116]	[0.07584 0.99712]	3125
[10296 11753]	[0.658944, 0.752192]	15625
[16124 76443]	[0.2063872, 0.9784704]	78125
[164833 354144]	[0.42197248 0.90660864]	390625

Other two-dimensional integer vectors with integer norms

All 2-dimensional integer vectors with integer 2-norms less than $100$ are:

Vector	Norm
[ 5 12]	13
[ 8 15]	17
[20 21]	29
[12 35]	37
[ 9 40]	41
[28 45]	53
[11 60]	61
[16 63]	65
[33 56]	65
[48 55]	73
[13 84]	85
[36 77]	85
[39 80]	89
[65 72]	97

Three-dimensional vectors

Generating mutually-orthogonal three-dimensional vectors

Given any two values $\alpha$ and $\beta$ it is always possible to construct three mutually orthogonal vectors as follows:

$\begin{pmatrix} \alpha \\ \beta \\ -\frac{\alpha \beta}{\alpha + \beta} \end{pmatrix}$, $\begin{pmatrix} \beta \\ -\frac{\alpha \beta}{\alpha + \beta} \\ \alpha \end{pmatrix}$, $\begin{pmatrix} -\frac{\alpha \beta}{\alpha + \beta} \\ \alpha \\ \beta \end{pmatrix}$

Therefore, if the first vector is normalized, so are the second and third. If normalized, the following, therefore, is a symmetric orthogonal matrix:

$\begin{pmatrix} \alpha & \beta & \frac{\alpha \beta}{\alpha + \beta} \\ \beta & -\frac{\alpha \beta}{\alpha + \beta} & -\alpha \\ \frac{\alpha \beta}{\alpha + \beta} & -\alpha & \beta \end{pmatrix}$.

Swapping any two columns will produce a non-symmetric matrix (assuming these three values are different). Any row or column can be multipled by $-1$.

All such integer vectors that have a 2-norm less than $100$ are:

$\left( \alpha, \beta, \frac{\alpha \beta}{\alpha + \beta}\right)$	Norm
[ 2  2  1]	3
[ 3  6  2]	7
[ 4 12  3]	13
[10 15  6]	19
[ 5 20  4]	21
[ 6 30  5]	31
[21 28 12]	37
[14 35 10]	39
[ 7 42  6]	43
[24 40 15]	49
[ 8 56  7]	57
[36 45 20]	61
[18 63 14]	67
[ 9 72  8]	73
[30 70 21]	79
[10 90  9]	91
[55 66 30]	91
[44 77 28]	93
[33 88 24]	97

If one vector is normalized, so are all, so one such set is:

$\begin{pmatrix} \frac{2}{3} \\ \frac{2}{3} \\ \frac{1}{3} \end{pmatrix}$, $\begin{pmatrix} \frac{2}{3} \\ -\frac{1}{3} \\ -\frac{2}{3} \end{pmatrix}$, $\begin{pmatrix} \frac{1}{3} \\ -\frac{2}{3} \\ \frac{2}{3} \end{pmatrix}$

No such triplet have terminating decimal representations when normalized. The following is a corresponding orthogonal matrix:

$\frac{1}{3}\begin{pmatrix} 2 & 2 & 1 \\ 2 & -1 & -2 \\ 1 & -2 & 2 \end{pmatrix}$.

Normalized three-dimensional vectors with terminating decimal representation

Vector	Normalized vector	Norm
[ 9 12  20]	[0.36  0.48  0.8]	25
[12 15  16]	[0.48  0.6   0.64]	25
[ 3 80  96]	[0.024 0.64  0.768]	125
[12 45 116]	[0.096 0.64  0.768]	125
[12 60 109]	[0.096 0.48  0.872]	125
[19 60 108]	[0.152 0.48  0.864]	125
[21 28 120]	[0.168 0.224 0.96]	125
[21 72 100]	[0.128 0.576 0.8]	125
[24 60 107]	[0.192 0.48  0.856]	125
[24 80  93]	[0.192 0.64  0.744]	125
[28 75  96]	[0.224 0.6   0.768]	125
[35 72  96]	[0.28  0.576 0.768]	125
[44 45 108]	[0.362 0.36  0.864]	125
[53 60  96]	[0.424 0.48  0.768]	125

You can view more in this text file with all such vectors with norms 25, 125, 625, 3125 and 15625. All such vectors have a norm equal to a power of five. The number of such sorted vectors with norm $5^{n + 1}$ is given by $a(n) = \frac{5^n - 1}{2}$, which is the sequence A125831.

Mutually orthogonal normalized three-dimensional vectors with terminating decimal representation

There are no triplets of mutually orthogonal normalized three-dimensional vectors with two of fewer digits beyond the decimal point.

Maximum three (3) digits after the decimal point

[0.36 0.48 0.8]  [ 0.864  0.152 -0.48]  [0.352 -0.864  0.36]
[0.36 0.48 0.8]  [-0.928  0.096  0.36]  [0.096 -0.872  0.48]
[0.48 0.6  0.64] [ 0.768 -0.64   0.024] [0.424  0.48  -0.768]
[0.48 0.6  0.64] [ 0.856 -0.48  -0.192] [0.192  0.64  -0.744]

Maximum four (4) digits after the decimal point

[0.36 0.48 0.8]     [ 0.6336 -0.7552  0.168]  [ 0.6848  0.4464 -0.576]
[0.36 0.48 0.8]     [-0.9024  0.3968  0.168]  [ 0.2368  0.7824 -0.576]
[0.48 0.6  0.64]    [ 0.6672  0.224  -0.7104] [ 0.5696 -0.768  0.2928]
[0.48 0.6  0.64]    [ 0.1216 -0.768   0.6288] [-0.8688  0.224  0.4416]
[0.224 0.6 0.768]   [ 0.7104 -0.64 0.2928]    [ 0.6672 0.48 -0.5696]
[0.224 0.6 0.768]   [ 0.8688 -0.48 0.1216]    [ 0.4416 0.64 -0.6288]
[0.168 0.576 0.8]   [ 0.6336 -0.6848  0.36]   [ 0.7552  0.4464 -0.48]
[0.168 0.576 0.8]   [ 0.9024  0.2368 -0.36]   [ 0.3968 -0.7824  0.48]
[0.024 0.64 0.768]  [ 0.9728 -0.192  0.1296]  [ 0.2304  0.744 -0.6272]
[0.024 0.64 0.768]  [ 0.9984  0.024 -0.0512]  [ 0.0512 -0.768  0.6384]
[0.096 0.36 0.928]  [ 0.9072  0.352 -0.2304]  [ 0.4096 -0.864  0.2928]
[0.096 0.36 0.928]  [ 0.9856  0.096 -0.1392]  [ 0.1392 -0.928  0.3456]
[0.096 0.48 0.872]  [ 0.8448  0.424 -0.3264]  [ 0.5264 -0.768  0.3648]
[0.096 0.48 0.872]  [ 0.9584  0.192 -0.2112]  [ 0.2688 -0.856  0.4416]
[0.152 0.48 0.864]  [ 0.9024 -0.424  0.0768]  [ 0.4032  0.768 -0.4976]
[0.192 0.48 0.856]  [ 0.9584  0.096 -0.2688]  [ 0.2112 -0.872  0.4416]
[0.192 0.64 0.744]  [ 0.8976  0.192 -0.3968]  [ 0.3968 -0.744  0.5376]
[0.352 0.36 0.864]  [ 0.4752 -0.864  0.1664]  [ 0.8064  0.352 -0.4752]
[0.352 0.36 0.864]  [ 0.9072  0.096 -0.4096]  [ 0.2304 -0.928  0.2928]
[0.424 0.48 0.768]  [ 0.8448  0.096 -0.5264]  [ 0.3264 -0.872  0.3648]
[0.152 0.48 0.864]  [-0.9792  0.192  0.0656]  [ 0.1344  0.856 -0.4992]
[0.192 0.48 0.856]  [ 0.0656  0.864 -0.4992]  [-0.9792  0.152  0.1344]
[0.192 0.64 0.744]  [-0.9728  0.024  0.2304]  [ 0.1296 -0.768  0.6272]
[0.424 0.48 0.768]  [ 0.0768 -0.864  0.4976]  [-0.9024  0.152  0.4032]
[0.168 0.224 0.96]  [ 0.2944 -0.9408  0.168]  [ 0.9408  0.2544 -0.224]
[0.168 0.224 0.96]  [-0.9856  0.0192  0.168]  [ 0.0192 -0.9744  0.224]
[0.28  0.576 0.768] [ 0.768  -0.6144  0.1808] [ 0.576   0.5392 -0.6144]
[0.28  0.576 0.768] [ 0.768   0.3456 -0.5392] [ 0.576  -0.7408  0.3456]

See this text file to see all triplets of mutually orthogonal 3-dimensional vectors with a finite decimal representation with no more than 3, 4, 5 or 6 digits after the decimal point.

Other three-dimensional integer vectors with integer norms

Vector	Norm
[1  2  2]	3
[2  3  6]	7
[1  4  8]	9
[4  4  7]	9
[2  6  9]	11
[6  6  7]	11
[3  4 12]	13
[2  5 14]	15
[2 10 11]	15
[1 12 12]	17
[8  9 12]	17
[6 10 15]	19
[1 12 12]	17
[8  9 12]	17
[6 10 15]	19

You can view more in this text file with all such vectors with at most two-digit entries.

Four-dimensional vectors

Generating mutually-orthogonal four-dimensional vectors

Given any one vector $\begin{pmatrix} \alpha \\ \beta \\ \gamma \\ \delta \end{pmatrix}$ in ${\bf R}^4$, it is always possible to construct four mutually orthogonal vectors as follows:

$\begin{pmatrix} \alpha \\ \beta \\ \gamma \\ \delta \end{pmatrix}$, $\begin{pmatrix} \beta \\ -\alpha \\ \delta \\ -\gamma \end{pmatrix}$, $\begin{pmatrix} \gamma \\ -\delta \\ -\alpha \\ \beta \end{pmatrix}$, $\begin{pmatrix} \delta \\ \gamma \\ -\beta \\ -\alpha \end{pmatrix}$

Therefore, if the first vector is normalized, then all four vectors will be normalize.

Other mutually-orthogonal four-dimensional vectors

While the above generates a set of orthogonal vectors, the entries are simply permutations and negations of the entries of the first vector. Here are other vectors that are not simple permutations.

[0.5  0.5  0.1  0.7 
 0.5  0.5 -0.1 -0.7 
 0.5 -0.5 -0.7  0.1 
 0.5 -0.5  0.7 -0.1]

[0.4  0.8  0.44  0.08 
 0.4  0.2 -0.8   0.4  
 0.8 -0.4  0.08 -0.44 
 0.2 -0.4  0.4   0.8 ]

[0.4   0.4   0.2   0.8  
 0.4  -0.4   0.8  -0.2  
 0.52  0.64 -0.08 -0.56 
 0.64 -0.52 -0.56  0.08]

[0.4   0.4   0.2   0.8  
 0.4   0.4  -0.8  -0.2  
 0.52  0.32  0.56 -0.56 
 0.64 -0.76 -0.08  0.08]

Normalized four-dimensional vectors with terminating decimal representation

These are all vectors in ${\bf R}^4$ that have norms of 1, 5, 10 or 25 (up to permutations and negation) where the norm is less than or equal to 25, and thus, these are the only integer vectors in ${\bf R}^4$ with such magnitudes that have normalized variations that have terminating decimal representations.

Vector	Normalized vector	Norm
[ 1  1  1  1]	[0.5  0.5  0.5  0.5]	2
[ 1  2  2  4]	[0.2  0.4  0.4  0.8]	5
[ 1  1  7  7]	[0.1  0.1  0.7  0.7]	10
[ 1  5  5  7]	[0.1  0.5  0.5  0.7]	10
[ 1  3  3  9]	[0.1  0.3  0.3  0.9]	10
[ 2  2 16 19]	[0.08 0.08 0.64 0.76]	25
[ 2  3  6 24]	[0.08 0.12 0.24 0.96]	25
[ 2  4 11 22]	[0.08 0.16 0.44 0.88]	25
[ 2  5 14 20]	[0.08 0.2  0.56 0.8]	25
[ 2  6 12 21]	[0.08 0.24 0.48 0.84]	25
[ 2  8 14 19]	[0.08 0.32 0.56 0.76]	25
[ 2 10 11 20]	[0.08 0.4  0.44 0.8]	25
[ 2 13 14 16]	[0.08 0.52 0.56 0.64]	25
[ 3  6 16 28]	[0.12 0.24 0.64 0.72]	25
[ 4  4  8 23]	[0.16 0.16 0.32 0.92]	25
[ 4  5 10 22]	[0.16 0.2  0.4  0.88]	25
[ 4 16 17 25]	[0.16 0.32 0.64 0.68]	25
[ 6 11 12 18]	[0.24 0.44 0.48 0.72]	25
[ 7  8 16 16]	[0.28 0.32 0.64 0.64]	25
[ 8 10 10 19]	[0.32 0.4  0.4  0.76]	25
[ 8 13 14 14]	[0.32 0.52 0.56 0.56]	25
[ 9 12 12 16]	[0.36 0.48 0.48 0.64]	25
[10 10 13 16]	[0.4  0.4  0.48 0.64]	25

You can view more in this text file with all such vectors with at most two-digit entries.

Other four-dimensional integer vectors with integer norms

All other integer vectors with integer norms less than or equal to 20 (up to permutations and signs) are are listed here grouped by the 2-norms.

Vector	Norm
[1  1  3  5]	6
[1  4  4  4]	7
[2  2  4  5]	7
[2  2  3  8]	9
[2  4  5  6]	9
[1  2  4 10]	11
[2  2  7  8]	11
[4  4  5  8]	11
[1  2  8 10]	13
[2  4  7 10]	13
[4  4  4 11]	13
[4  5  8  8]	13
[4  6  6  9]	13
[1  1  5 13]	14
[1  5  7 11]	14
[3  3  3 13]	14
[3  5  9  9]	14
[5  5  5 11]	14
[1  4  8 12]	15
[2  3  4 14]	15
[2  4  6 13]	15
[2  6  8 11]	15
[3  4 10 10]	15
[4  4  7 12]	15
[4  8  8  9]	15
[5  6  8 10]	15
[1  4  4 16]	17
[2  2  5 16]	17
[2  4 10 13]	17
[2  5  8 14]	17
[2  8 10 11]	17
[3  6 10 12]	17
[5  8 10 10]	17
[1  3  5 17]	18
[1  7  7 15]	18
[1  9 11 11]	18
[3  5 11 13]	18
[5  5  7 15]	18
[5  7  9 13]	18
[1  2 10 16]	19
[1  8 10 14]	19
[2  2  8 17]	19
[3  8 12 12]	19
[4  5  8 16]	19
[4  7 10 14]	19
[6  6  8 15]	19
[6  9 10 12]	19
[8  8  8 13]	19

Five-dimensional vectors

Normalized five-dimensional vectors with terminating decimal representation

Each of these integer vectors, when divided by the norm, has a terminating decimal representation. The vectors are grouped by the 2-norm of the integer vector, with the integer vector first and the normalized vector second.

      Norm 4
      [ 1  1  1  2 3]  [0.25 0.25 0.25 0.5  0.75]
 
      Norm 5
      [ 2  2  2  2 3]  [0.4  0.4  0.4  0.4  0.6]
 
      Norm 10
      [ 1  1  1  4 9]  [0.1  0.1  0.1  0.4  0.9]
      [ 1  1  3  5 8]  [0.1  0.1  0.3  0.5  0.8]
      [ 1  3  4  5 7]  [0.1  0.3  0.4  0.5  0.7]
      [ 3  3  3  3 8]  [0.3  0.3  0.3  0.3  0.8]
      [ 3  4  5  5 5]  [0.3  0.4  0.5  0.5  0.5]
 
      Norm 20
      [ 1  1  1  6 19]  [0.05 0.05 0.05 0.3  0.95]
      [ 1  1  2 13 15]  [0.05 0.05 0.1  0.65 0.75]
      [ 1  1  3 10 17]  [0.05 0.05 0.15 0.5  0.85]
      [ 1  1  5  7 18]  [0.05 0.05 0.25 0.35 0.9 ]
      [ 1  1  9 11 14]  [0.05 0.05 0.45 0.55 0.7 ]
      [ 1  2  3  5 19]  [0.05 0.1  0.15 0.25 0.95]
      [ 1  2  5  9 17]  [0.05 0.1  0.25 0.45 0.85]
      [ 1  2  7 11 15]  [0.05 0.1  0.35 0.55 0.75]
      [ 1  3  5 13 14]  [0.05 0.15 0.25 0.65 0.7 ]
      [ 1  3 10 11 13]  [0.05 0.15 0.5  0.55 0.65]
      [ 1  5  5  5 18]  [0.05 0.25 0.25 0.25 0.9 ]
      [ 1  5  6  7 17]  [0.05 0.25 0.3  0.35 0.85]
      [ 1  5  6 13 13]  [0.05 0.25 0.3  0.65 0.65]
      [ 1  5  7 10 15]  [0.05 0.25 0.35 0.5  0.75]
      [ 1  6 11 11 11]  [0.05 0.3  0.55 0.55 0.55]
      [ 1  7  9 10 13]  [0.05 0.35 0.45 0.5  0.65]
      [ 2  3  7  7 17]  [0.1  0.15 0.35 0.35 0.85]
      [ 2  3  7 13 13]  [0.1  0.15 0.35 0.65 0.65]
      [ 2  3  9  9 15]  [0.1  0.15 0.45 0.45 0.75]
      [ 2  5  5 11 15]  [0.1  0.25 0.25 0.55 0.75]
      [ 2  5  9 11 13]  [0.1  0.25 0.45 0.55 0.65]
      [ 3  3  3  7 18]  [0.15 0.15 0.15 0.35 0.9 ]
      [ 3  3  6 11 15]  [0.15 0.15 0.3  0.55 0.75]
      [ 3  5  7 11 14]  [0.15 0.25 0.35 0.55 0.7 ]
      [ 3  6  7  9 15]  [0.15 0.3  0.35 0.45 0.75]
      [ 3  7 10 11 11]  [0.15 0.35 0.5  0.55 0.55]
      [ 5  5  5  6 17]  [0.25 0.25 0.25 0.3  0.85]
      [ 5  5  9 10 13]  [0.25 0.25 0.45 0.5  0.65]
      [ 5  6  7 11 13]  [0.25 0.3  0.35 0.55 0.65]
      [ 5  7  7  9 14]  [0.25 0.35 0.35 0.45 0.7 ]
      [ 6  9  9  9 11]  [0.3  0.45 0.45 0.45 0.55]
      [ 7  7  9 10 11]  [0.35 0.35 0.45 0.5  0.55]

      Norm 25
      [ 1  2  6 10 22]  [0.04 0.08 0.24 0.4  0.88]
      [ 1  2 10 14 18]  [0.04 0.08 0.4  0.56 0.72]
      [ 1  4  4  4 24]  [0.04 0.16 0.16 0.16 0.96]
      [ 1  4  8 12 20]  [0.04 0.16 0.32 0.48 0.8 ]
      [ 1  6 14 14 14]  [0.04 0.24 0.56 0.56 0.56]
      [ 1 10 10 10 18]  [0.04 0.4  0.4  0.4  0.72]
      [ 2  2  2 17 18]  [0.08 0.08 0.08 0.68 0.72]
      [ 2  2  4  5 24]  [0.08 0.08 0.16 0.2  0.96]
      [ 2  2 14 14 15]  [0.08 0.08 0.56 0.56 0.6 ]
      [ 2  3  4 14 20]  [0.08 0.12 0.16 0.56 0.8 ]
      [ 2  3  8  8 22]  [0.08 0.12 0.32 0.32 0.88]
      [ 2  3 10 16 16]  [0.08 0.12 0.4  0.64 0.64]
      [ 2  3 12 12 18]  [0.08 0.12 0.48 0.48 0.72]
      [ 2  4  5 16 18]  [0.08 0.16 0.2  0.64 0.72]
      [ 2  4  6 13 20]  [0.08 0.16 0.24 0.52 0.8 ]
      [ 2  4  8 10 21]  [0.08 0.16 0.32 0.4  0.84]
      [ 2  4 10 12 19]  [0.08 0.16 0.4  0.48 0.76]
      [ 2  5 12 14 16]  [0.08 0.2  0.48 0.56 0.64]
      [ 2  6  6 15 18]  [0.08 0.24 0.24 0.6  0.72]
      [ 2  6  8 11 20]  [0.08 0.24 0.32 0.44 0.8 ]
      [ 2  6 10 14 17]  [0.08 0.24 0.4  0.56 0.68]
      [ 2  8  8 13 18]  [0.08 0.32 0.32 0.52 0.72]
      [ 2 10 10 14 15]  [0.08 0.4  0.4  0.56 0.6 ]
      [ 2 10 11 12 16]  [0.08 0.4  0.44 0.48 0.64]
      [ 3  4  4 10 22]  [0.12 0.16 0.16 0.4  0.88]
      [ 3  4 10 10 20]  [0.12 0.16 0.4  0.4  0.8 ]
      [ 3  6  6 12 20]  [0.12 0.24 0.24 0.48 0.8 ]
      [ 3  8 10 14 16]  [0.12 0.32 0.4  0.56 0.64]
      [ 4  4  6 14 19]  [0.16 0.16 0.24 0.56 0.76]
      [ 4  4  7 12 20]  [0.16 0.16 0.28 0.48 0.8 ]
      [ 4  4  9 16 16]  [0.16 0.16 0.36 0.64 0.64]
      [ 4  4 10 13 18]  [0.16 0.16 0.4  0.52 0.72]
      [ 4  5  6  8 22]  [0.16 0.2  0.24 0.32 0.88]
      [ 4  5  8 14 18]  [0.16 0.2  0.32 0.56 0.72]
      [ 4  6 11 14 16]  [0.16 0.24 0.44 0.56 0.64]
      [ 4  8  8  9 20]  [0.16 0.32 0.32 0.36 0.8 ]
      [ 4  8  8 15 16]  [0.16 0.32 0.32 0.6  0.64]
      [ 4  8 10 11 18]  [0.16 0.32 0.4  0.44 0.72]
      [ 4 10 12 13 14]  [0.16 0.4  0.48 0.52 0.56]
      [ 5  6  8 10 20]  [0.2  0.24 0.32 0.4  0.8 ]
      [ 5  8 12 14 14]  [0.2  0.32 0.48 0.56 0.56]
      [ 5 10 10 12 16]  [0.2  0.4  0.4  0.48 0.64]
      [ 6  8  8 10 19]  [0.24 0.32 0.32 0.4  0.76]
      [ 6  8 10 13 16]  [0.24 0.32 0.4  0.52 0.64]
      [ 6 10 10 10 17]  [0.24 0.4  0.4  0.4  0.68]
      [ 7 12 12 12 12]  [0.28 0.48 0.48 0.48 0.48]
      [ 8  8  8 12 17]  [0.32 0.32 0.32 0.48 0.68]
      [ 8 10 11 12 14]  [0.32 0.4  0.44 0.48 0.56]

You can view more in this text file with all such vectors with a 2-norm less than or equal to 100.

Other five-dimensional integer vectors with integer norms

These are all such integer vectors that have norm less than or equal to 19. These integer vectors are grouped by the 2-norms.

Vector	Norm
[1  1  3  3  4]	6
[1  2  2  2  6]	7
[2  2  3  4  4]	7
[2  3  4  4  6]	9
[1  2  2  6  6]	9
[1  2  4  6  8]	11
[2  2  2  3 10]	11
[2  2  4  4  9]	11
[2  3  6  6  6]	11
[2  4  4  6  7]	11
[3  4  4  4  8]	11
[1  1  5  6  9]	12
[1  2  3  3 11]	12
[1  2  3  7  9]	12
[1  3  3  5 10]	12
[1  3  6  7  7]	12
[2  3  5  5  9]	12
[3  5  5  6  7]	12
[1  2  2  4 12]	13
[1  2  6  8  8]	13
[1  4  4  6 10]	13
[2  2  2  6 11]	13
[2  2  4  8  9]	13
[2  2  5  6 10]	13
[2  4  6  7  8]	13
[3  4  4  8  8]	13
[5  6  6  6  6]	13
[1  1  1  7 12]	14
[1  1  3  4 13]	14
[1  1  3  8 11]	14
[1  1  5  5 12]	14
[1  1  7  8  9]	14
[1  3  4  7 11]	14
[1  4  7  7  9]	14
[1  5  5  8  9]	14
[3  3  3  5 12]	14
[3  3  4  9  9]	14
[3  4  5  5 11]	14
[3  5  7  7  8]	14
[4  5  5  7  9]	14
[2  2  3  8 12]	15
[2  2  6  9 10]	15
[2  4  5  6 12]	15
[2  6  6  7 10]	15
[3  4  6  8 10]	15
[4  4  6  6 11]	15
[5  6  6  8  8]	15
[1  4  8  8 12]	17
[2  2  2  9 14]	17
[2  2  3  4 16]	17
[2  2  4 11 12]	17
[2  2  6  7 14]	17
[2  2  9 10 10]	17
[2  3  4  8 14]	17
[2  4  5 10 12]	17
[2  4  6  8 13]	17
[2  6  7 10 10]	17
[2  6  8  8 11]	17
[3  4  8 10 10]	17
[3  6  6  8 12]	17
[4  4  4  4 15]	17
[4  4  5  6 14]	17
[4  4  6 10 11]	17
[4  4  7  8 12]	17
[4  8  8  8  9]	17
[5  6  8  8 10]	17
[6  6  6  9 10]	17
[1  1  3 12 13]	18
[1  1  4  9 15]	18
[1  3  3  4 17]	18
[1  3  3  7 16]	18
[1  3  5  8 15]	18
[1  3  7 11 12]	18
[1  3  8  9 13]	18
[1  7  7  9 12]	18
[3  3  4 11 13]	18
[3  3  5  5 16]	18
[3  3  8 11 11]	18
[3  4  5  7 15]	18
[3  4  7  9 13]	18
[3  5  5 11 12]	18
[3  7  8  9 11]	18
[4  5  9  9 11]	18
[5  5  7  9 12]	18
[7  7  8  9  9]	18
[1  2  4  4 18]	19
[1  2  4 12 14]	19
[1  2  6  8 16]	19
[1  4 10 10 12]	19
[1  6  6 12 12]	19
[1  6  8  8 14]	19
[2  2  2  5 18]	19
[2  2  4  9 16]	19
[2  2  6 11 14]	19
[2  2  8  8 15]	19
[2  4  4  6 17]	19
[2  4  4 10 15]	19
[2  4  6  7 16]	19
[2  4  8  9 14]	19
[2  5  6 10 14]	19
[2  6 10 10 11]	19
[2  7  8 10 12]	19
[3  4  4  8 16]	19
[4  4  4 12 13]	19
[4  4  8 11 12]	19
[4  6  7  8 14]	19
[4  8  9 10 10]	19
[5  6 10 10 10]	19
[5  8  8  8 12]	19
[6  6  8  9 12]	19

Mutually-orthogonal five-dimensional vectors

These are the only nine matrices that are symmetric and orthogonal where each entry has either one digit beyond the decimal point or two digits where the least significant digit is a five (5):

                      0.1   0.1   0.1   0.4   0.9
                      0.1   0.1  -0.9   0.4  -0.1
                      0.1  -0.9   0.1   0.4  -0.1
                      0.4   0.4   0.4   0.6  -0.4
                      0.9  -0.1  -0.1  -0.4   0.1

                      0.1   0.3   0.4   0.5   0.7
                      0.3   0.7   0.4  -0.1  -0.5
                      0.4   0.4  -0.6  -0.4   0.4
                      0.5  -0.1  -0.4   0.7  -0.3
                      0.7  -0.5   0.4  -0.3   0.1

                      0.4   0.4   0.4   0.4   0.6
                      0.4   0.4   0.4  -0.6  -0.4
                      0.4   0.4  -0.6   0.4  -0.4
                      0.4  -0.6   0.4   0.4  -0.4
                      0.6  -0.4  -0.4  -0.4   0.4

                      0.05  0.05  0.05  0.3   0.95
                      0.05  0.05 -0.95  0.3  -0.05
                      0.05 -0.95  0.05  0.3  -0.05
                      0.3   0.3   0.3   0.8  -0.3
                      0.95 -0.05 -0.05 -0.3   0.05

                      0.05  0.25  0.3   0.35  0.85
                      0.25  0.85  0.3  -0.05 -0.35
                      0.3   0.3  -0.8  -0.3   0.3
                      0.35 -0.05 -0.3   0.85 -0.25
                      0.85 -0.35  0.3  -0.25  0.05

                      0.05  0.1   0.15  0.25  0.95
                      0.1  -0.8   0.3   0.5  -0.1
                      0.15  0.3  -0.55  0.75 -0.15
                      0.25  0.5   0.75  0.25 -0.25
                      0.95 -0.1  -0.15 -0.25  0.05

                      0.05  0.1   0.35  0.55  0.75
                      0.1   0.8  -0.1  -0.5   0.3
                      0.35 -0.1   0.85 -0.35 -0.15
                      0.55 -0.5  -0.35 -0.35  0.45
                      0.75  0.3  -0.15  0.45 -0.35

                      0.05  0.25  0.3   0.65  0.65
                      0.25 -0.75 -0.5   0.25  0.25
                      0.3  -0.5   0.8  -0.1  -0.1
                      0.65  0.25 -0.1   0.45 -0.55
                      0.65  0.25 -0.1  -0.55  0.45

                      0.15  0.3   0.35  0.45  0.75
                      0.3  -0.8   0.3  -0.3   0.3
                      0.35  0.3   0.75 -0.15 -0.45
                      0.45 -0.3  -0.15  0.75 -0.35
                      0.75  0.3  -0.45 -0.35  0.15

These are the only nine non-symmetric orthogonal matrices where each entry has either one digit beyond the decimal point or two digits where the least significant digit is a five (5):

                      0.05  0.05  0.05  0.3   0.95
                      0.05 -0.15  0.65  0.7  -0.25
                      0.45  0.85  0.25 -0.1  -0.05
                      0.55 -0.05 -0.65  0.5  -0.15
                      0.7  -0.5   0.3  -0.4   0.1

                      0.05  0.05  0.05  0.3   0.95
                      0.05 -0.55  0.65 -0.5   0.15
                      0.15 -0.05 -0.65 -0.7   0.25
                      0.5  -0.7  -0.3   0.4  -0.1
                      0.85  0.45  0.25 -0.1  -0.05  

                      0.05  0.05  0.1   0.65  0.75
                      0.05 -0.55 -0.3  -0.55  0.55
                      0.25 -0.75  0.5   0.25 -0.25
                      0.35  0.35  0.7  -0.45  0.25
                      0.9   0.1  -0.4   0.1  -0.1

                      0.05  0.05  0.1   0.65  0.75
                      0.25 -0.75  0.5   0.25 -0.25
                      0.3  -0.5  -0.4  -0.5   0.5
                      0.35  0.35  0.7  -0.45  0.25
                      0.85  0.25 -0.3   0.25 -0.25

                      0.05  0.05  0.15  0.5   0.85
                      0.1  -0.3  -0.1  -0.8   0.5
                      0.25  0.85  0.35 -0.3   0.05
                      0.45  0.25 -0.85  0.1   0.05
                      0.85 -0.35  0.35  0.1  -0.15

                      0.05  0.05  0.25  0.35  0.9
                      0.1  -0.3  -0.7  -0.5   0.4
                      0.15  0.35 -0.65  0.65 -0.1
                      0.25  0.85  0.05 -0.45  0.1
                      0.95 -0.25  0.15  0.05 -0.1

                      0.05  0.05  0.45  0.55  0.7
                      0.1   0.5  -0.3   0.7  -0.4
                      0.35  0.55 -0.45 -0.35  0.5
                      0.55 -0.65 -0.45  0.25  0.1
                      0.75  0.15  0.55 -0.15 -0.3

                      0.05  0.1   0.15  0.25  0.95
                      0.1  -0.8   0.3   0.5  -0.1
                      0.25  0.5   0.75  0.25 -0.25
                      0.45 -0.3   0.35 -0.75  0.15
                      0.85  0.1  -0.45  0.25 -0.05

                      0.05  0.1   0.15  0.25  0.95
                      0.15  0.3  -0.55  0.75 -0.15
                      0.25  0.5   0.75  0.25 -0.25
                      0.65 -0.7   0.15  0.25 -0.05
                      0.7   0.4  -0.3  -0.5   0.1

                      0.05  0.1   0.25  0.45  0.85
                      0.1   0.8   0.5   0.1  -0.3
                      0.35 -0.5   0.75 -0.25 -0.05
                      0.55  0.3  -0.25 -0.65  0.35
                      0.75 -0.1  -0.25  0.55 -0.25

                      0.05  0.1   0.25  0.45  0.85
                      0.15  0.3   0.75  0.35 -0.45
                      0.25  0.5   0.25 -0.75  0.25
                      0.65 -0.7   0.25 -0.15  0.05
                      0.7   0.4  -0.5   0.3  -0.1

                      0.05  0.15  0.25  0.65  0.7
                      0.25  0.55 -0.75  0.25 -0.1
                      0.3  -0.7  -0.5  -0.1   0.4
                      0.65 -0.25  0.25  0.45 -0.5
                      0.65  0.35  0.25 -0.55  0.3

                      0.05  0.1   0.35  0.55  0.75
                      0.25  0.5   0.75 -0.25 -0.25
                      0.3  -0.4   0.1  -0.7   0.5
                      0.35  0.7  -0.55 -0.15  0.25
                      0.85 -0.3  -0.05  0.35 -0.25

                      0.05  0.1   0.35  0.55  0.75
                      0.35  0.3   0.65  0.25 -0.55
                      0.45  0.5  -0.65  0.35 -0.05
                      0.5  -0.8  -0.1   0.3  -0.1
                      0.65  0.1   0.15 -0.65  0.35

                      0.05  0.25  0.25  0.25  0.9
                      0.3  -0.5  -0.5  -0.5   0.4
                      0.55 -0.25 -0.25  0.75 -0.1
                      0.55 -0.25  0.75 -0.25 -0.1
                      0.55  0.75 -0.25 -0.25 -0.1

                      0.05  0.25  0.35  0.5   0.75
                      0.35 -0.25 -0.55  0.7  -0.15
                      0.45 -0.75  0.15 -0.3   0.35
                      0.5   0.5  -0.5  -0.4   0.3
                      0.65  0.25  0.55  0.1  -0.45

                      0.1   0.1   0.1   0.4   0.9
                      0.15  0.15  0.15 -0.9   0.35
                      0.35 -0.65 -0.65 -0.1   0.15
                      0.65 -0.35  0.65  0.1  -0.15
                      0.65  0.65 -0.35  0.1  -0.15

                      0.1   0.15  0.35  0.65  0.65
                      0.3  -0.35  0.25 -0.65  0.55
                      0.4   0.5  -0.7  -0.1   0.3
                      0.5   0.55  0.55 -0.15 -0.35
                      0.7  -0.55 -0.15  0.35 -0.25

                      0.25  0.25  0.25  0.3   0.85
                      0.25  0.25 -0.75 -0.5   0.25
                      0.25 -0.75  0.25 -0.5   0.25
                      0.5   0.5   0.5  -0.4  -0.3
                      0.75 -0.25 -0.25  0.5  -0.25

                      0.3   0.3   0.3   0.3   0.8
                      0.45  0.45  0.45 -0.55 -0.3
                      0.45  0.45 -0.55  0.45 -0.3
                      0.45 -0.55  0.45  0.45 -0.3
                      0.55 -0.45 -0.45 -0.45  0.3

Six-dimensional vectors

Normalized six-dimensional vectors with terminating decimal representation

Each of these integer vectors, when divided by the norm, has a terminating decimal representation. The vectors are grouped by the 2-norm of the integer vector, with the integer vector first and the normalized vector second.

             Norm 5
             [1 1 1  2  3  3    0.2 0.2 0.2 0.4 0.6 0.6 

             Norm 10
             [1 1 1 5 6 6]  [0.1 0.1 0.1 0.5 0.6 0.6]
             [1 1 2 2 3 9]  [0.1 0.1 0.2 0.2 0.3 0.9]
             [1 1 2 3 6 7]  [0.1 0.1 0.2 0.3 0.6 0.7]
             [1 1 3 3 4 8]  [0.1 0.1 0.3 0.3 0.4 0.8]
             [1 2 3 5 5 6]  [0.1 0.2 0.3 0.5 0.5 0.6]
             [1 3 3 3 6 6]  [0.1 0.3 0.3 0.3 0.6 0.6]
             [1 3 3 4 4 7]  [0.1 0.3 0.3 0.4 0.4 0.7]
             [2 2 2 4 6 6]  [0.2 0.2 0.2 0.4 0.6 0.6]
             [2 2 3 3 5 7]  [0.2 0.2 0.3 0.3 0.5 0.7]
             [3 3 4 4 5 5]  [0.3 0.3 0.4 0.4 0.5 0.5]

             Norm 20
             [2  2  2 10 12 12]  [0.1 0.1 0.1 0.5 0.6 0.6]
             [2  2  4  4  6 18]  [0.1 0.1 0.2 0.2 0.3 0.9]
             [2  2  4  6 12 14]  [0.1 0.1 0.2 0.3 0.6 0.7]
             [2  2  6  6  8 16]  [0.1 0.1 0.3 0.3 0.4 0.8]
             [2  4  6 10 10 12]  [0.1 0.2 0.3 0.5 0.5 0.6]
             [2  6  6  6 12 12]  [0.1 0.3 0.3 0.3 0.6 0.6]
             [2  6  6  8  8 14]  [0.1 0.3 0.3 0.4 0.4 0.7]
             [4  4  4  8 12 12]  [0.2 0.2 0.2 0.4 0.6 0.6]
             [4  4  6  6 10 14]  [0.2 0.2 0.3 0.3 0.5 0.7]
             [6  6  8  8 10 10]  [0.3 0.3 0.4 0.4 0.5 0.5]

             [1  1  1  3  8 18]  [0.05 0.05 0.05 0.15 0.4  0.9 ]
             [1  1  2  5 12 15]  [0.05 0.05 0.1  0.25 0.6  0.75]
             [1  1  2  9 12 13]  [0.05 0.05 0.1  0.45 0.6  0.65]
             [1  1  3  4  7 18]  [0.05 0.05 0.15 0.2  0.35 0.9 ]
             [1  1  3  6  8 17]  [0.05 0.05 0.15 0.3  0.4  0.85]
             [1  1  3  7 12 14]  [0.05 0.05 0.15 0.35 0.6  0.7 ]
             [1  1  3  8 10 15]  [0.05 0.05 0.15 0.4  0.5  0.75]
             [1  1  4  6 11 15]  [0.05 0.05 0.2  0.3  0.55 0.75]
             [1  1  5  6  9 16]  [0.05 0.05 0.25 0.3  0.45 0.8 ]
             [1  1  6  7 12 13]  [0.05 0.05 0.3  0.35 0.6  0.65]
             [1  2  3  3  4 19]  [0.05 0.1  0.15 0.15 0.2  0.95]
             [1  2  3  3 11 16]  [0.05 0.1  0.15 0.15 0.55 0.8 ]
             [1  2  3  4  9 17]  [0.05 0.1  0.15 0.2  0.45 0.85]
             [1  2  3  7  9 16]  [0.05 0.1  0.15 0.35 0.45 0.8 ]
             [1  2  3 11 11 12]  [0.05 0.1  0.15 0.55 0.55 0.6 ]
             [1  2  5  8  9 15]  [0.05 0.1  0.25 0.4  0.45 0.75]
             [1  2  7  9 11 12]  [0.05 0.1  0.35 0.45 0.55 0.6 ]
             [1  2  8  9  9 13]  [0.05 0.1  0.4  0.45 0.45 0.65]
             [1  3  3  4 13 14]  [0.05 0.15 0.15 0.2  0.65 0.7 ]
             [1  3  3  5 10 16]  [0.05 0.15 0.15 0.25 0.5  0.8 ]
             [1  3  3  8 11 14]  [0.05 0.15 0.15 0.4  0.55 0.7 ]
             [1  3  4  5  5 18]  [0.05 0.15 0.2  0.25 0.25 0.9 ]
             [1  3  4  6  7 17]  [0.05 0.15 0.2  0.3  0.35 0.85]
             [1  3  4  6 13 13]  [0.05 0.15 0.2  0.3  0.65 0.65]
             [1  3  4  7 10 15]  [0.05 0.15 0.2  0.35 0.5  0.75]
             [1  3  5  5 12 14]  [0.05 0.15 0.25 0.25 0.6  0.7 ]
             [1  3  5 10 11 12]  [0.05 0.15 0.25 0.5  0.55 0.6 ]
             [1  3  6  7  7 16]  [0.05 0.15 0.3  0.35 0.35 0.8 ]
             [1  3  6  8 11 13]  [0.05 0.15 0.3  0.4  0.55 0.65]
             [1  3  7  8  9 14]  [0.05 0.15 0.35 0.4  0.45 0.7 ]
             [1  4  5  9  9 14]  [0.05 0.2  0.25 0.45 0.45 0.7 ]
             [1  4  9  9 10 11]  [0.05 0.2  0.45 0.45 0.5  0.55]
             [1  5  5  6 12 13]  [0.05 0.25 0.25 0.3  0.6  0.65]
             [1  5  6  7  8 15]  [0.05 0.25 0.3  0.35 0.4  0.75]
             [1  5  7  9 10 12]  [0.05 0.25 0.35 0.45 0.5  0.6 ]
             [1  6  7  7 11 12]  [0.05 0.3  0.35 0.35 0.55 0.6 ]
             [1  6  7  8  9 13]  [0.05 0.3  0.35 0.4  0.45 0.65]
             [2  3  3  3 12 15]  [0.1  0.15 0.15 0.15 0.6  0.75]
             [2  3  3  5  8 17]  [0.1  0.15 0.15 0.25 0.4  0.85]
             [2  3  4  5 11 15]  [0.1  0.15 0.2  0.25 0.55 0.75]
             [2  3  4  9 11 13]  [0.1  0.15 0.2  0.45 0.55 0.65]
             [2  3  5  5  9 16]  [0.1  0.15 0.25 0.25 0.45 0.8 ]
             [2  3  5  7 12 13]  [0.1  0.15 0.25 0.35 0.6  0.65]
             [2  3  7  7  8 15]  [0.1  0.15 0.35 0.35 0.4  0.75]
             [2  3  8  9 11 11]  [0.1  0.15 0.4  0.45 0.55 0.55]
             [2  3  9  9  9 12]  [0.1  0.15 0.45 0.45 0.45 0.6 ]
             [2  4  5  7  9 15]  [0.1  0.2  0.25 0.35 0.45 0.75]
             [2  4  7  9  9 13]  [0.1  0.2  0.35 0.45 0.45 0.65]
             [2  5  5  9 11 12]  [0.1  0.25 0.25 0.45 0.55 0.6 ]
             [2  7  8  9  9 11]  [0.1  0.35 0.4  0.45 0.45 0.55]
             [3  3  3  6  9 16]  [0.15 0.15 0.15 0.3  0.45 0.8 ]
             [3  3  4  7 11 14]  [0.15 0.15 0.2  0.35 0.55 0.7 ]
             [3  3  6  9 11 12]  [0.15 0.15 0.3  0.45 0.55 0.6 ]
             [3  3  7  8 10 13]  [0.15 0.15 0.35 0.4  0.5  0.65]
             [3  4  5  5  6 17]  [0.15 0.2  0.25 0.25 0.3  0.85]
             [3  4  5  5 10 15]  [0.15 0.2  0.25 0.25 0.5  0.75]
             [3  4  5  9 10 13]  [0.15 0.2  0.25 0.45 0.5  0.65]
             [3  4  6  7 11 13]  [0.15 0.2  0.3  0.35 0.55 0.65]
             [3  4  7  7  9 14]  [0.15 0.2  0.35 0.35 0.45 0.7 ]
             [3  5  5  6  7 16]  [0.15 0.25 0.25 0.3  0.35 0.8 ]
             [3  5  5  8  9 14]  [0.15 0.25 0.25 0.4  0.45 0.7 ]
             [3  5  8  9 10 11]  [0.15 0.25 0.4  0.45 0.5  0.55]
             [3  6  7  8 11 11]  [0.15 0.3  0.35 0.4  0.55 0.55]
             [3  6  7  9  9 12]  [0.15 0.3  0.35 0.45 0.45 0.6 ]
             [3  7  7  7 10 12]  [0.15 0.35 0.35 0.35 0.5  0.6 ]
             [4  5  6  7  7 15]  [0.2  0.25 0.3  0.35 0.35 0.75]
             [4  5  6  9 11 11]  [0.2  0.25 0.3  0.45 0.55 0.55]
             [4  6  7  7  9 13]  [0.2  0.3  0.35 0.35 0.45 0.65]
             [5  5  5  6  8 15]  [0.25 0.25 0.25 0.3  0.4  0.75]
             [5  5  5  9 10 12]  [0.25 0.25 0.25 0.45 0.5  0.6 ]
             [5  5  6  7 11 12]  [0.25 0.25 0.3  0.35 0.55 0.6 ]
             [5  5  6  8  9 13]  [0.25 0.25 0.3  0.4  0.45 0.65]
             [5  7  8  9  9 10]  [0.25 0.35 0.4  0.45 0.45 0.5 ]
             [6  7  7  8  9 11]  [0.3  0.35 0.35 0.4  0.45 0.55]

You can view more in this text file with all such vectors with a 2-norm less than or equal to 100.

Other six-dimensional integer vectors with integer norms

These are all such integer vectors that have norm less than or equal to 19. These integer vectors are grouped by the 2-norms.

  [1  1  1  1  1  2]     3
  [1  1  1  1  4  4]     6
  [1  1  1  2  2  5]     6
  [1  2  2  3  3  3]     6
  [2  2  2  2  2  4]     6
  [1  1  1  1  3  6]     7
  [1  1  2  3  3  5]     7
  [2  3  3  3  3  3]     7
  [1  1  1  2  5  7]     9
  [1  1  2  5  5  5]     9
  [1  1  3  3  5  6]     9
  [1  2  2  2  2  8]     9
  [1  2  3  3  3  7]     9
  [1  4  4  4  4  4]     9
  [2  2  2  2  4  7]     9
  [2  2  4  4  4  5]     9
  [2  3  3  3  5  5]     9
  [3  3  3  3  3  6]     9
  [1  1  1  1  6  9]    11
  [1  1  1  3  3 10]    11
  [1  1  2  3  5  9]    11
  [1  1  3  5  6  7]    11
  [1  2  3  3  7  7]    11
  [1  3  5  5  5  6]    11
  [1  4  4  4  6  6]    11
  [2  2  2  3  6  8]    11
  [2  2  4  5  6  6]    11
  [2  3  3  3  3  9]    11
  [2  3  3  5  5  7]    11
  [3  3  3  3  6  7]    11
  [1  1  1  2  4 11]    12
  [1  1  1  4  5 10]    12
  [1  1  2  5  7  8]    12
  [1  1  3  4  6  9]    12
  [1  2  4  5  7  7]    12
  [1  2  5  5  5  8]    12
  [1  3  3  3  4 10]    12
  [1  3  3  5  6  8]    12
  [2  2  2  2  8  8]    12
  [2  2  2  4  4 10]    12
  [2  3  3  3  7  8]    12
  [2  3  3  4  5  9]    12
  [2  4  4  6  6  6]    12
  [2  4  5  5  5  7]    12
  [3  3  4  5  6  7]    12
  [4  4  4  4  4  8]    12
  [1  1  1  2  9  9]    13
  [1  1  1  3  6 11]    13
  [1  1  1  6  7  9]    13
  [1  1  3  3  7 10]    13
  [1  1  5  5  6  9]    13
  [1  2  3  3  5 11]    13
  [1  2  3  5  7  9]    13
  [1  3  3  5  5 10]    13
  [1  3  5  6  7  7]    13
  [1  4  4  6  6  8]    13
  [2  2  2  2  3 12]    13
  [2  2  3  4  6 10]    13
  [2  2  5  6  6  8]    13
  [2  3  3  7  7  7]    13
  [2  3  5  5  5  9]    13
  [2  4  4  4  6  9]    13
  [3  3  3  5  6  9]    13
  [3  4  6  6  6  6]    13
  [3  5  5  5  6  7]    13
  [4  4  4  6  6  7]    13
  [1  1  1  6  6 11]    14
  [1  1  2  3  9 10]    14
  [1  1  3  4  5 12]    14
  [1  1  3  6  7 10]    14
  [1  1  4  4  9  9]    14
  [1  2  2  3  3 13]    14
  [1  2  2  5  9  9]    14
  [1  2  3  5  6 11]    14
  [1  2  5  6  7  9]    14
  [1  3  3  7  8  8]    14
  [1  3  4  5  8  9]    14
  [1  3  5  5  6 10]    14
  [1  5  6  6  7  7]    14
  [2  2  2  2  6 12]    14
  [2  2  3  3  7 11]    14
  [2  2  3  7  7  9]    14
  [2  2  4  6  6 10]    14
  [2  3  3  5  7 10]    14
  [2  3  6  7  7  7]    14
  [2  5  5  5  6  9]    14
  [3  3  3  3  4 12]    14
  [3  3  4  4  5 11]    14
  [3  3  4  7  7  8]    14
  [3  3  5  5  8  8]    14
  [3  3  5  6  6  9]    14
  [3  4  4  5  7  9]    14
  [4  6  6  6  6  6]    14
  [5  5  5  6  6  7]    14
  [1  1  1  1  5 14]    15
  [1  1  1  1 10 11]    15
  [1  1  1  2  7 13]    15
  [1  1  2  5  5 13]    15
  [1  1  2  7  7 11]    15
  [1  1  3  3  3 14]    15
  [1  1  3  3  6 13]    15
  [1  1  5  6  9  9]    15
  [1  1  5  7  7 10]    15
  [1  2  2  2  4 14]    15
  [1  2  2  4 10 10]    15
  [1  2  2  6  6 12]    15
  [1  2  3  3  9 11]    15
  [1  2  3  7  9  9]    15
  [1  2  5  5  7 11]    15
  [1  3  3  5  9 10]    15
  [1  3  3  6  7 11]    15
  [1  3  6  7  7  9]    15
  [1  4  4  8  8  8]    15
  [1  4  6  6  6 10]    15
  [1  5  5  5  7 10]    15
  [2  2  2  7  8 10]    15
  [2  2  4  4  4 13]    15
  [2  2  4  4  8 11]    15
  [2  2  5  8  8  8]    15
  [2  2  6  6  8  9]    15
  [2  3  3  3  5 13]    15
  [2  3  4  4  6 12]    15
  [2  3  5  5  9  9]    15
  [2  4  4  5  8 10]    15
  [2  5  5  5  5 11]    15
  [2  5  7  7  7  7]    15
  [2  6  6  6  7  8]    15
  [3  3  3  6  9  9]    15
  [3  3  3  7  7 10]    15
  [3  3  5  5  6 11]    15
  [3  4  6  6  8  8]    15
  [3  5  5  6  7  9]    15
  [4  4  4  7  8  8]    15
  [5  5  5  5  5 10]    15
  [1  1  1  3  9 14]    17
  [1  1  1  5  6 15]    17
  [1  1  1  6  9 13]    17
  [1  1  2  3  7 15]    17
  [1  1  2  9  9 11]    17
  [1  1  3  3 10 13]    17
  [1  1  3  6 11 11]    17
  [1  1  5  9  9 10]    17
  [1  1  6  7  9 11]    17
  [1  2  2  6 10 12]    17
  [1  2  3  5  5 15]    17
  [1  2  3  5  9 13]    17
  [1  2  4  6  6 14]    17
  [1  3  3  3  6 15]    17
  [1  3  3  5  7 14]    17
  [1  3  3  7 10 11]    17
  [1  3  5  6  7 13]    17
  [1  3  6  9  9  9]    17
  [1  3  7  7  9 10]    17
  [1  4  6  6 10 10]    17
  [1  5  5  6  9 11]    17
  [1  6  6  6  6 12]    17
  [2  2  2  4  6 15]    17
  [2  2  3  8  8 12]    17
  [2  2  6  8  9 10]    17
  [2  3  3  5 11 11]    17
  [2  3  3  7  7 13]    17
  [2  3  4  4 10 12]    17
  [2  3  5  7  9 11]    17
  [2  4  5  6  8 12]    17
  [2  5  7  7  9  9]    17
  [2  6  6  7  8 10]    17
  [3  3  3  9  9 10]    17
  [3  3  5  5  5 14]    17
  [3  3  5  5 10 11]    17
  [3  4  4  4  6 14]    17
  [3  4  6  8  8 10]    17
  [3  5  5  5  6 13]    17
  [3  5  5  7  9 10]    17
  [3  5  6  7  7 11]    17
  [4  4  4  4  9 12]    17
  [4  4  4  6  6 13]    17
  [4  4  6  6  8 11]    17
  [5  6  6  8  8  8]    17
  [5  6  7  7  7  9]    17
  [6  6  6  6  8  9]    17
  [1  1  1  1  8 16]    18
  [1  1  1  2 11 14]    18
  [1  1  1  4  4 17]    18
  [1  1  1  4  7 16]    18
  [1  1  1  5 10 14]    18
  [1  1  1 10 10 11]    18
  [1  1  2  2  5 17]    18
  [1  1  2  7 10 13]    18
  [1  1  3  5 12 12]    18
  [1  1  3  6  9 14]    18
  [1  1  4  4 11 13]    18
  [1  1  4  5  5 16]    18
  [1  1  4  8 11 11]    18
  [1  1  4  9  9 12]    18
  [1  1  5  6  6 15]    18
  [1  1  5  8  8 13]    18
  [1  1  6  6  9 13]    18
  [1  2  2  3  9 15]    18
  [1  2  2  5 11 13]    18
  [1  2  3  6  7 15]    18
  [1  2  5  5 10 13]    18
  [1  2  5  7  7 14]    18
  [1  2  6  9  9 11]    18
  [1  2  7  7 10 11]    18
  [1  3  3  3 10 14]    18
  [1  3  3  4  8 15]    18
  [1  3  3  6 10 13]    18
  [1  3  5  8  9 12]    18
  [1  3  6  6 11 11]    18
  [1  4  4  7 11 11]    18
  [1  4  5  7  8 13]    18
  [1  4  8  9  9  9]    18
  [1  5  6  9  9 10]    18
  [1  5  7  7 10 10]    18
  [1  5  7  8  8 11]    18
  [1  6  6  7  9 11]    18
  [2  2  2  4 10 14]    18
  [2  2  3  3  3 17]    18
  [2  2  4 10 10 10]    18
  [2  2  5  7 11 11]    18
  [2  2  6  6 10 12]    18
  [2  2  7  7  7 13]    18
  [2  3  3  5  9 14]    18
  [2  3  3  9 10 11]    18
  [2  3  5  5  6 15]    18
  [2  3  5  6  9 13]    18
  [2  3  7  9  9 10]    18
  [2  4  4  4  4 16]    18
  [2  4  6  6  6 14]    18
  [2  5  5  5  7 14]    18
  [2  5  5  7 10 11]    18
  [2  8  8  8 8   8]    18
  [3  3  3  3 12 12]    18
  [3  3  3  4  5 16]    18
  [3  3  3  6  6 15]    18
  [3  3  3  8  8 13]    18
  [3  3  4  4  7 15]    18
  [3  3  4  5 11 12]    18
  [3  3  5  6  7 14]    18
  [3  3  5  9 10 10]    18
  [3  3  6  7 10 11]    18
  [3  3  7  7  8 12]    18
  [3  4  4  9  9 11]    18
  [3  4  5  7  9 12]    18
  [3  5  6  6  7 13]    18
  [3  5  8  8  9  9]    18
  [3  6  6  9  9  9]    18
  [3  6  7  7  9 10]    18
  [4  4  4  4  8 14]    18
  [4  4  5  5 11 11]    18
  [4  4  5  7  7 13]    18
  [4  4  7  9  9  9]    18
  [4  4  8  8  8 10]    18
  [4  5  5  5  8 13]    18
  [4  5  7  7  8 11]    18
  [4  6  6  6 10 10]    18
  [5  5  5  7 10 10]    18
  [5  5  5  8  8 11]    18
  [5  5  6  6  9 11]    18
  [6  6  6  6  6 12]    18
  [7  7  7  7  8  8]    18
  [1  1  1  3  5 18]    19
  [1  1  1  9  9 14]    19
  [1  1  2  3 11 15]    19
  [1  1  2  7  9 15]    19
  [1  1  3  5  6 17]    19
  [1  1  3  5 10 15]    19
  [1  1  3  9 10 13]    19
  [1  1  6  7  7 15]    19
  [1  1  6  9 11 11]    19
  [1  2  2  8 12 12]    19
  [1  2  3  3  7 17]    19
  [1  2  3  3 13 13]    19
  [1  2  5  5  9 15]    19
  [1  2  5  9  9 13]    19
  [1  3  3  3  3 18]    19
  [1  3  3  5 11 14]    19
  [1  3  3  6  9 15]    19
  [1  3  3 10 11 11]    19
  [1  3  5  6 11 13]    19
  [1  3  5  7  9 14]    19
  [1  3  7  9 10 11]    19
  [1  4  4  6  6 16]    19
  [1  4  6  8 10 12]    19
  [1  5  5  6  7 15]    19
  [1  5  6  7  9 13]    19
  [1  6  9  9  9  9]    19
  [1  7  7  9  9 10]    19
  [2  2  2  3  4 18]    19
  [2  2  2  3 12 14]    19
  [2  2  2  6 12 13]    19
  [2  2  3 10 10 12]    19
  [2  2  4  7 12 12]    19
  [2  2  5  6  6 16]    19
  [2  2  8  8  9 12]    19
  [2  3  3  5  5 17]    19
  [2  3  3  7 11 13]    19
  [2  3  4  6 10 14]    19
  [2  3  5  7  7 15]    19
  [2  3  5  9 11 11]    19
  [2  3  7  7  9 13]    19
  [2  4  4  6  8 15]    19
  [2  4  4  9 10 12]    19
  [2  4  6  6 10 13]    19
  [2  5  6  6  8 14]    19
  [2  5  7  9  9 11]    19
  [2  6  6  8 10 11]    19
  [2  6  7  8  8 12]    19
  [3  3  3  3  6 17]    19
  [3  3  3  3 10 15]    19
  [3  3  5  7 10 13]    19
  [3  3  7  7  7 14]    19
  [3  3  9  9  9 10]    19
  [3  4  6 10 10 10]    19
  [3  4  8  8  8 12]    19
  [3  5  5  5  9 14]    19
  [3  5  5  9 10 11]    19
  [3  5  6  7 11 11]    19
  [3  6  6  6 10 12]    19
  [3  6  7  7  7 13]    19
  [4  4  4  5 12 12]    19
  [4  4  4  6  9 14]    19
  [4  4  6  7 10 12]    19
  [4  6  8  8  9 10]    19
  [5  5  5  5  6 15]    19
  [5  5  5  6  9 13]    19
  [5  5  7  9  9 10]    19
  [5  6  6  8 10 10]    19
  [5  6  7  7  9 11]    19