cp's OEIS Frontend

A046738 Period of Fibonacci 3-step sequence A000073 mod n.

%I A046738 #52 Sep 01 2025 18:24:16
%S A046738 1,4,13,8,31,52,48,16,39,124,110,104,168,48,403,32,96,156,360,248,624,
%T A046738 220,553,208,155,168,117,48,140,1612,331,64,1430,96,1488,312,469,360,
%U A046738 2184,496,560,624,308,440,1209,2212,46,416,336,620,1248,168
%N A046738 Period of Fibonacci 3-step sequence A000073 mod n.
%C A046738 Could also be called the tribonacci Pisano periods. [_Carl R. White_, Oct 05 2009]
%C A046738 Klaska notes that n=208919=59*3541 satisfies a(n) = a(n^2). - _Michel Marcus_, Mar 03 2016
%C A046738 39, 78, 273, 546 also satisfy a(n) = a(n^2). - _Michel Marcus_, Mar 07 2016
%H A046738 T. D. Noe [1..1000] + Jean-François Alcover [1001..2000] + Zhong Ziqian [2001..20000], <a href="/A046738/b046738.txt">Table of n, a(n) for n = 1..20000</a>
%H A046738 Jirí Klaška, <a href="http://dml.cz/dmlcz/137497">A search for Tribonacci-Wieferich primes</a>, Acta Mathematica Universitatis Ostraviensis, vol. 16 (2008), issue 1, pp. 15-20.
%H A046738 Jirí Klaška, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-4/Klaska.pdf">On Tribonacci-Wieferich primes</a>, Fibonacci Quart. 46/47 (2008/2009), no. 4, 290-297.
%H A046738 Jirí Klaška, <a href="http://dx.doi.org/10.1007/s10114-010-8433-8">Tribonacci partition formulas modulo m</a>, Acta Mathematica Sinica, English Series, March 2010, Volume 26, Issue 3, pp 465-476.
%H A046738 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).
%F A046738 a(3^k) = 13*3^(k-1) for k > 0. If a(p) != a(p^2) for p prime, then a(p^k) = p^(k-1)*a(p) for k > 0 [Waddill, 1978]. - _Chai Wah Wu_, Feb 25 2022
%F A046738 Let the prime factorization of n be p1^e1*...*pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)) [Waddill, 1978]. - _Avery Diep_, Aug 26 2025
%p A046738 a:= proc(n) local f, k, l; l:= ifactors(n)[2];
%p A046738       if nops(l)<>1 then ilcm(seq(a(i[1]^i[2]), i=l))
%p A046738     else f:= [0, 0, 1];
%p A046738          for k do f:=[f[2], f[3], f[1]+f[2]+f[3] mod n];
%p A046738                   if f=[0, 0, 1] then break fi
%p A046738          od; k
%p A046738       fi
%p A046738     end:
%p A046738 seq(a(n), n=1..100);  # _Alois P. Heinz_, Aug 27 2023
%t A046738 Table[a = {0, 1, 1}; a = a0 = Mod[a, n]; k = 0; While[k++; s = a[[3]] + a[[2]] + a[[1]]; a = RotateLeft[a]; a[[-1]] = Mod[s, n]; a != a0]; k, {n, 100}] (* _T. D. Noe_, Aug 28 2012 *)
%o A046738 (Python)
%o A046738 from itertools import count
%o A046738 def A046738(n):
%o A046738     a = b = (0,0,1%n)
%o A046738     for m in count(1):
%o A046738         b = b[1:] + (sum(b) % n,)
%o A046738         if a == b:
%o A046738             return m # _Chai Wah Wu_, Feb 27 2022
%Y A046738 Cf. A106302.
%Y A046738 Cf. A001175.
%K A046738 nonn,changed
%O A046738 1,2
%A A046738 _David W. Wilson_