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.
Geometric sequences represent a non-orthogonal basis for infinite signals, and are intimately connected with the z-transform. Thus, they play a significant roll in engineering design. In this page, "nice" means a finite number of digits after the decimal point.
Here we present geometric sequences with:
The only geometric sequence that has both nice 1- and 2-norms as well as a nice normalization is any sequence with the ratio $r = \frac{3}{5}z = 0.6z$ where $|z| = 1$: if $x_z = (1, 0.6z, 0.36z^2, 0.216z^3, 0.1296z^4, 0.07776z^5, \ldots)$ then $\|x_z\|_1 = \frac{1}{1 - \frac{3}{5}} = \frac{5}{2} = 2.5$ and $\|x_z\|_2 = \sqrt{\frac{1}{1 - \frac{9}{25}}} = \frac{5}{4} = 1.25$. this sequence normalized under the 1-norm is $\hat{x_z} = (0.4, 0.24z, 0.144z^2, 0.0864z^3, 0.05184z^4, 0.031104z^5, \ldots)$ and normalized under the 2-norm is twice this sequence: $\hat{x_z} = (0.8, 0.48z, 0.288z^2, 0.1728z^3, 0.10368z^4, 0.062208z^5, \ldots)$.
If you are interested, $\langle x_1, x_z \rangle = \frac{25}{25 - 9z}$ if $|z| = 1$, and if $z = \cos(\theta) + \sin(\theta)j$, then $\langle x_1, x_z \rangle = \frac{25}{2} \frac{9z^* - 25}{225 \cos(\theta) - 353}$. This traces out a circle in the complex plane centered at $\frac{625}{544} \approx 1.148897$ with a radius of $\frac{225}{544} \approx 0.413603$. The circle is not, however, traced linearly, and the maximum imaginary points of the circle $\frac{625}{544} \pm \frac{225}{544}j$ is achieved when $\theta = \pm \tan^{-1}\left(\frac{272}{225}\right) \approx \pm 0.8797 \approx \pm 50.40^\circ$. Of course, they are closest to orothgonal with $z = -1$, where the inner product is $\sum_{k = 0}^\infty -\frac{3^2}{5^2} = \frac{25}{34}$; however, the angle is not close to $90^\circ$: $\cos^{-1}\left(\frac{25}{16}\right) \approx 1.08084 \approx 61.9275^\circ$.
| Ratio $r$ | 1-norm | 2-norm |
|---|---|---|
| $\frac{3}{5} = 0.6$ | $\frac{5}{2} = 2.5$ | $\frac{5}{4} = 1.25$ |
| $\frac{12}{13}$ | $\frac{13}{1} = 13$ | $\frac{13}{5} = 2.6$ |
| $\frac{15}{17}$ | $\frac{17}{2} = 8.5$ | $\frac{17}{8} = 2.125$ |
| $\frac{21}{29}$ | $\frac{29}{8} = 3.625$ | $\frac{29}{20} = 1.45$ |
| $\frac{9}{41}$ | $\frac{41}{32} = 1.28125$ | $\frac{41}{40} = 1.025$ |
| $\frac{63}{65}$ | $\frac{65}{2} = 32.5$ | $\frac{65}{16} = 4.0625$ |
| $\frac{39}{89}$ | $\frac{89}{50} = 1.78$ | $\frac{89}{80} = 1.1125$ |
| $\frac{99}{101}$ | $\frac{101}{2} = 50.5$ | $\frac{101}{20} = 5.05$ |
| $\frac{255}{257}$ | $\frac{257}{2} = 128.5$ | $\frac{257}{32} = 8.03125$ |
| $\frac{231}{281}$ | $\frac{281}{50} = 5.62$ | $\frac{281}{160} = 1.75625$ |
| $\frac{312}{313}$ | $\frac{313}{1} = 313$ | $\frac{313}{25} = 12.52$ |
| $\frac{399}{401}$ | $\frac{401}{2} = 200.5$ | $\frac{401}{40} = 10.025$ |
| $\frac{621}{629}$ | $\frac{629}{8} = 78.625$ | $\frac{629}{100} = 6.29$ |
| $\frac{609}{641}$ | $\frac{641}{32} = 20.03125$ | $\frac{641}{200} = 3.205$ |
| $\frac{561}{689}$ | $\frac{689}{128} = 5.3828125$ | $\frac{689}{400} = 1.7225$ |
| $\frac{369}{881}$ | $\frac{881}{512} = 1.720703125$ | $\frac{881}{800} = 1.10125$ |
| $\frac{1023}{1025}$ | $\frac{1025}{2} = 512.5$ | $\frac{1025}{64} = 16.015625$ |
| $\frac{999}{1049}$ | $\frac{1049}{50} = 20.98$ | $\frac{1049}{320} = 3.278125$ |
| $\frac{1599}{1601}$ | $\frac{1601}{2} = 800.5$ | $\frac{1601}{80} = 20.0125$ |
| $\frac{399}{1649}$ | $\frac{1649}{1250} = 1.3192$ | $\frac{1649}{1600} = 1.030625$ |
| $\frac{2499}{2501}$ | $\frac{2501}{2} = 1250.5$ | $\frac{2501}{100} = 25.01$ |
| $\frac{4095}{4097}$ | $\frac{4097}{2} = 2048.5$ | $\frac{4097}{128} = 32.0078125$ |
| $\frac{4071}{4121}$ | $\frac{4121}{50} = 82.42$ | $\frac{4121}{640} = 6.4390625$ |
| $\frac{3471}{4721}$ | $\frac{4721}{1250} = 3.7768$ | $\frac{4721}{3200} = 1.4753125$ |
| $\frac{6399}{6401}$ | $\frac{6401}{2} = 3200.5$ | $\frac{6401}{160} = 40.00625$ |
| $\frac{7812}{7813}$ | $\frac{7813}{1} = 7813$ | $\frac{7813}{125} = 62.504$ |
| $\frac{9999}{10001}$ | $\frac{10001}{2} = 5000.5$ | $\frac{10001}{200} = 50.005$ |
| $\frac{15621}{15629}$ | $\frac{15629}{8} = 1953.625$ | $\frac{15629}{500} = 31.258$ |
| $\frac{15609}{15641}$ | $\frac{15641}{32} = 488.78125$ | $\frac{15641}{1000} = 15.641$ |
| $\frac{15561}{15689}$ | $\frac{15689}{128} = 122.5703125$ | $\frac{15689}{2000} = 7.8445$ |
| $\frac{15369}{15881}$ | $\frac{15881}{512} = 31.017578125$ | $\frac{15881}{4000} = 3.97025$ |
| $\frac{16383}{16385}$ | $\frac{16385}{2} = 8192.5$ | $\frac{16385}{256} = 64.00390625$ |
| $\frac{16359}{16409}$ | $\frac{16409}{50} = 328.18$ | $\frac{16409}{1280} = 12.81953125$ |
| $\frac{14601}{16649}$ | $\frac{16649}{2048} = 8.12939453125$ | $\frac{16649}{8000} = 2.081125$ |
| $\frac{15759}{17009}$ | $\frac{17009}{1250} = 13.6072$ | $\frac{17009}{6400} = 2.65765625$ |
In the academic setting, it is occasionally convenient to have geometric sequences that have 1-, 2- and infinity-norms such that the resulting normalized geometric sequences have terminating decimal representation. This is difficult, because while the infinity-norm is trivial, and the 1-norm will be rational if and only if the ratio is rational (thus, it is straight-forward to find ratios such that the 1-norm has a numerator comprised of prime factors including only 2 or 5), but the 2-norm includes a square root, so the series of the squares of that geometric sequence must be the square of a rational number which, in lowest terms, has a numerator that has prime factors that consist of powers of 2 and 5 only.
Fortunately, there are such ratios, but they are relatively rare and are associated with Pythagorean triplets. Unfortunately, there is only one Pythagorean triplet for each power of five, and no Pythagorean triplet includes a hypotenuse that is even, let alone a numerator that has only 2 and 5 as factors.
The only ratios are of the form $\frac{a_n}{5^n}$ and $\frac{b_n}{5^n}$ where $a_n$ is the smallest leg and $b_n$ is the largest leg in a right-angled triangle with relatively prime sides and hypotenuse $5^n$.
Of course, if you're only looking for geometric sequences where the 2-norm is rational, then you need only choose any Pythagorean triplet and divide the two sides by the hypotenuse to get two ratios that are less than one and that have a rational 2-norm.
Note: Some Pythagorean triplets produce ratios that result in both 1- and 2-norms that have finite decimal representations, but when normalizing the corresponding geometric sequence, the resulting sequence does not have a finite decimal representation. For example, each of the following ratios produce 1- and 2-norms that have finite decimal representations, but do not result in normalized geometric sequences that, too, have finite decimal representations: 12/13, 15/17, 21/29, 9/41, 63/65, 39/89, 99/101, 255/257, 231/281.
x = (1, 0.6, 0.36, 0.216, 0.1296, 0.07776, 0.046656, ...)
||x||1 = 5/2 = 2.5
Normalized x = (0.4, 0.24, 0.144, 0.0864, 0.05184, 0.031104, 0.0186624, ...)
||x||2 = 5/4 = 1.25
Normalized x = (0.8, 0.48, 0.288, 0.1728, 0.10368, 0.062208, 0.0373248, ...)
x = (1, 0.8, 0.64, 0.512, 0.4096, 0.32768, 0.262144, ...)
||x||1 = 5
Normalized x = (0.2, 0.16, 0.128, 0.1024, 0.08192, 0.065536, 0.0524288, ...)
||x||2 = 5/3
Normalized x = (0.6, 0.48, 0.384, 0.3072, 0.24576, 0.196608, 0.1572864, ...)
x = (1, 0.28, 0.0784, 0.021952, 0.00614656, 0.0017210368, 0.000481890304, ...)
||x||1 = 25/18
Normalized x = (0.72, 0.2016, 0.056448, 0.01580544, 0.0044255232, 0.001239146496, 0.00034696101888, ...)
||x||2 = 25/24
Normalized x = (0.96, 0.2688, 0.075264, 0.02107392, 0.0059006976, 0.001652195328, 0.00046261469184, ...)
x = (1, 0.96, 0.9216, 0.884736, 0.84934656, 0.8153726976, 0.782757789696, ...)
||x||1 = 25
Normalized x = (0.04, 0.0384, 0.036864, 0.03538944, 0.0339738624, 0.032614907904, 0.03131031158784, ...)
||x||2 = 25/7
Normalized x = (0.28, 0.2688, 0.258048, 0.24772608, 0.2378170368, 0.228304355328, 0.21917218111488, ...)
x = (1, 0.352, 0.123904, 0.043614208, 0.015352201216, 0.005403974828032, 0.001902199139467264, ...)
||x||1 = 125/81
Normalized x = (0.648, 0.228096, 0.080289792, 0.028262006784, 0.009948226387968, 0.003501775688564736, 0.001232625042374787072, ...)
||x||2 = 125/117
Normalized x = (0.936, 0.329472, 0.115974144, 0.040822898688, 0.014369660338176, 0.005058120439037952, 0.001780458394541359104, ...)
x = (1, 0.936, 0.876096, 0.820025856, 0.767544201216, 0.718421372338176, 0.672442404508532736, ...)
||x||1 = 125/8
Normalized x = (0.064, 0.059904, 0.056070144, 0.052481654784, 0.049122828877824, 0.045978967829643264, 0.043036313888546095104, ...)
||x||2 = 125/44
Normalized x = (0.352, 0.329472, 0.308385792, 0.288649101312, 0.270175558828032, 0.252884323063037952, 0.236699726387003523072, ...)
x = (1, 0.5376, 0.28901376, 0.155373797376, 0.0835289534693376, 0.04490516538511589376, 0.024141016911038304485376, ...)
||x||1 = 625/289
Normalized x = (0.4624, 0.24858624, 0.133639962624, 0.0718448439066624, 0.03862378808422170624, 0.020764148474077589274624, 0.0111628062196641119940378624, ...)
||x||2 = 625/527
Normalized x = (0.8432, 0.45330432, 0.243696402432, 0.1310111859474432, 0.07043161356534546432, 0.037864035452729721618432, 0.0203557054593874983420690432, ...)
x = (1, 0.8432, 0.71098624, 0.599503597568, 0.5055014334693376, 0.42623880870134546432, 0.359404563496974495514624, ...)
||x||1 = 625/98
Normalized x = (0.1568, 0.13221376, 0.111482642432, 0.0940021640986624, 0.07926262476799213568, 0.066834245204370968805376, 0.0563546355563256008966930432, ...)
||x||2 = 625/336
Normalized x = (0.5376, 0.45330432, 0.382226202624, 0.3222931340525568, 0.27175757063311589376, 0.229145983557843321618432, 0.1932158933359734887886618624, ...)
x = (1, 0.07584, 0.0057517056, 0.000436209352704, 0.00003308211730907136, 0.0000025089477767199719424, 0.000000190278599386442672111616, ...)
||x||1 = 3125/2888
Normalized x = (0.92416, 0.0700882944, 0.005315496247296, 0.00040312723539492864, 0.0000305731695323513880576, 0.000002318669177333529270288384, 0.00000017584787040897485985867104256, ...)
||x||2 = 3125/3116
Normalized x = (0.99712, 0.0756215808, 0.005735140687872, 0.00043495306976821248, 0.0000329868408112212344832, 0.000002501722007123018423205888, 0.00000018973059702020971721593454592, ...)
x = (1, 0.99712, 0.9942482944, 0.991384859312128, 0.98852967091730907136, 0.9856827054650672212344832, 0.982843939273327827637327888384, ...)
||x||1 = 3125/9
Normalized x = (0.00288, 0.0028717056, 0.002863435087872, 0.00285518839481892864, 0.0028469654522418501255168, 0.002838766191739393597155311616, 0.00283059054510718414359550431854592, ...)
||x||2 = 3125/237
Normalized x = (0.07584, 0.0756215808, 0.075403790647296, 0.07518662773023178752, 0.0749700902423687199719424, 0.074754176382470698058423205888, 0.07453888435448918244801494705504256, ...)
x = (1, 0.658944, 0.434207195136, 0.286118225991696384, 0.188535888307872382058496, 0.124234592385142658923153588224, 0.081863639244635444241458518038675456, ...)
||x||1 = 15625/5329
Normalized x = (0.341056, 0.224736804864, 0.148088969144303616, 0.097582337683824001941504, 0.064301295922729723135342411776, 0.042370953140507214681695070185324544, 0.027920085346218386071214876328198496321536, ...)
||x||2 = 15625/11753
Normalized x = (0.752192, 0.495652405248, 0.326607178523738112, 0.215215840645146086473728, 0.141815186898075142805344223232, 0.093448266515365226900724743833387008, 0.061577174530700824074871165600547368599552, ...)
x = (1, 0.752192, 0.565792804864, 0.425584821476261888, 0.320121498035872382058496, 0.240792829850598918805344223232, 0.181122440270981701934029481961324544, ...)
||x||1 = 15625/3872
Normalized x = (0.247808, 0.186399195136, 0.140207983387738112, 0.105463323440389505941504, 0.079328668185273463253151776768, 0.059670389579617216871314741270675456, 0.044883589678671433592867977865871912599552, ...)
||x||2 = 15625/10296
Normalized x = (0.658944, 0.495652405248, 0.372825774008303616, 0.280436564602853913526272, 0.210942140401749890923153588224, 0.158668990473073053953268743833387008, 0.119349545281921766599217122961523040321536, ...)
x = (1, 0.2063872, 0.04259567632384, 0.008791202368583630848, 0.0018143916414853435365523456, 0.00037446721058956389354713626181632, 0.000077285239085390441210291521094737199104, ...)
||x||1 = 78125/62001
Normalized x = (0.7936128, 0.16379152367616, 0.033804473955256369152, 0.0069768107270982873114476544, 0.00143992443089577964300520933818368, 0.000297181971504173452336844740721582800896, 0.0000613345549892261471421348428722534538450829312, ...)
||x||2 = 78125/76443
Normalized x = (0.9784704, 0.20194376613888, 0.041678608450858254336, 0.0086019312980689727092948992, 0.00177532851520082068434778822017024, 0.000366405081332454818744623836953919356928, 0.0000756213188019776191672104287621759451021705216, ...)
x = (1, 0.9784704, 0.95740432367616, 0.936791791549141745664, 0.9166230389938053435365523456, 0.89688851161348431201234778822017024, 0.877578860713850640168446745278905262800896, ...)
||x||1 = 78125/1682
Normalized x = (0.0215296, 0.02106607632384, 0.020612532127018254336, 0.0201687525553364021274476544, 0.01973452738032103152420455737982976, 0.019309650899633671843901042941264977199104, 0.0188939218396249187425705910471567187459981705216, ...)
||x||2 = 78125/16124
Normalized x = (0.2063872, 0.20194376613888, 0.197595997631416369152, 0.1933418348408110272907051008, 0.18917926247342230219754713626181632, 0.185106308624074509400154825436953919356928, 0.1811210438419216348425732521072264762547410829312, ...)
x = (1, 0.42197248, 0.1780607738773504, 0.075136746343744764116992, 0.03170563919380091060146212438016, 0.0133789072005933708727572642507645779968, 0.005645530651124242178737147233910470873463128064, ...)
||x||1 = 390625/225792
Normalized x = (0.57802752, 0.2439117061226496, 0.102924027533605635883008, 0.04343110714994385351552987561984, 0.0183267319932075397287048601293954220032, 0.007733376549469128694020117016854107123336871936, 0.00326327208135333091845482994749212938102012572627632128, ...)
||x||2 = 390625/354144
Normalized x = (0.90660864, 0.3825638962102272, 0.161431436042292172947456, 0.06811962341672741310322691801088, 0.0287446064298225399911531585958077005824, 0.012129432861816163159966076392506293017852772352, 0.00511828686569406367269552197121533388035001862424887296, ...)
x = (1, 0.90660864, 0.8219392261226496, 0.745177203957707827052544, 0.67558409143910011060146212438016, 0.6124903743452381940962411585958077005824, 0.555289065298227289625649225906569529126536871936, ...)
||x||1 = 390625/36481
Normalized x = (0.09339136, 0.0846694138773504, 0.076762022164941772947456, 0.06959311251860771645108187561984, 0.0630937170938619165052209657843522994176, 0.057201309047010904470591932689238171455863128064, 0.05185920100133025216725327209036176125968689056024887296, ...)
||x||2 = 390625/164833
Normalized x = (0.42197248, 0.3825638962102272, 0.346835733656255235883008, 0.31444427279349978689677308198912, 0.2850778945131038426387732642507645779968, 0.258454082238588536953512240370746293017852772352, 0.23431670400077490900701347546587539249795699766227632128, ...)