This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351889 #42 Feb 16 2025 08:34:03 %S A351889 1,1,1,1,3,1,1,8,4,1,1,6,13,5,1,1,20,8,26,6,1,1,24,31,10,104,7,1,1,16, %T A351889 52,312,12,728,8,1,1,12,48,130,781,14,364,9,1,1,24,16,342,312,208,16, %U A351889 80,10,1,1,60,39,20,2801,728,9372,18,91,11,1,1,10,124,78,24,342,728,195312 %N A351889 Table T(n,k) read by downward antidiagonals: period of n-step Fibonacci numbers mod k, n >= 1, k >= 1. %H A351889 Chai Wah Wu, <a href="/A351889/b351889.txt">Table of n, a(n) for n = 1..226</a> %H A351889 M. E. Waddill, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/16-4/waddill.pdf">Some properties of a generalized Fibonacci sequence modulo m</a>, The Fibonacci Quarterly, vol. 16, no. 4, pp. 344-353 (1978). %H A351889 Marcellus E. Waddill, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/30-3/waddill.pdf">Some Properties of the Tetranacci Sequence Modulo m</a>, The Fibonacci Quarterly, vol. 30, no. 3, 232-238 (1992). %H A351889 D. D. Wall, <a href="https://doi.org/10.1080/00029890.1960.11989541">Fibonacci Series Modulo m</a>, The American Mathematical Monthly, vol. 67, no. 6, pp. 525-532, (1960). %H A351889 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a> %H A351889 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisanoPeriod.html">Pisano Period</a> %F A351889 T(1,k) = T(n,1) = 1. %F A351889 T(2,k) = A001175(k). %F A351889 T(3,k) = A046738(k). %F A351889 T(4,k) = A106295(k) for k not a multiple of 563. %F A351889 T(5,k) = A106303(k). %F A351889 T(n,2) = n + 1 for n > 1. %F A351889 T(n,3) = A337212(n). %F A351889 T(n,n) = A351657(n). %F A351889 T(n,p_1^e_1*...*p_m^e_m) = lcm(T(n,p_1^e_1),...,T(n,p_m^e_m)) for p_i distinct primes. %F A351889 Conjecture 1: T(n,2^m) = (n+1)*2^(m-1) for n > 1. %F A351889 Conjecture 2: For p prime, if T(n,p) != T(n,p^2) then T(n,p^k) = p^(k-1)T(n,p). %F A351889 Conjecture 2 is true for n = 2, n = 3 and n = 4 (see [Wall, 1960], [Waddill, 1978] and [Waddill, 1992] resp.). It is easy to show that T(n,4) != n+1 for all n, and thus Conjecture 2 implies Conjecture 1. %F A351889 Conjecture 3: T(p^m,p^k) = (p^(pm)-1)*p^(k-1)/(p^m-1) for p prime and k, m > 0. %e A351889 Table T(n,k) starts: %e A351889 1 1 1 1 1 1 1 1 1 1 %e A351889 1 3 8 6 20 24 16 12 24 60 %e A351889 1 4 13 8 31 52 48 16 39 124 %e A351889 1 5 26 10 312 130 342 20 78 1560 %e A351889 1 6 104 12 781 312 2801 24 312 4686 %e A351889 1 7 728 14 208 728 342 28 2184 1456 %e A351889 1 8 364 16 9372 728 137257 32 1092 18744 %e A351889 1 9 80 18 195312 720 13680 36 240 585936 %e A351889 1 10 91 20 488281 910 5764800 40 273 4882810 %e A351889 1 11 8744 22 19344 96184 19152 44 26232 212784 %e A351889 1 12 3851 24 406224 46212 109531200 48 11553 406224 %o A351889 (Python) %o A351889 from functools import lru_cache %o A351889 from math import lcm %o A351889 from itertools import count %o A351889 from sympy import factorint %o A351889 @lru_cache(maxsize=None) %o A351889 def A351889_T(n,k): # computes the period of the n-step Fibonacci sequence mod k %o A351889 if len(fs := factorint(k)) <= 1: %o A351889 a = b = (0,)*(n-1)+(1 % k,) %o A351889 s = 1 % k %o A351889 for m in count(1): %o A351889 b, s = b[1:] + (s,), (s + s - b[0]) % k %o A351889 if a == b: %o A351889 return m %o A351889 else: %o A351889 return lcm(*(A351889_T(n,p**e) for p, e in fs.items())) %Y A351889 Cf. A000045, A001175, A046738, A106295, A106303, A337212, A351657 (diagonal). %K A351889 nonn,tabl %O A351889 1,5 %A A351889 _Chai Wah Wu_, Feb 24 2022