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A161553 Table which contains in row n the fundamental Pisano period of the Fibonacci sequence (mod n).

%I A161553 #27 Feb 16 2025 08:33:10
%S A161553 0,0,1,1,0,1,1,2,0,2,2,1,0,1,1,2,3,1,0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,
%T A161553 2,2,4,1,0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2,5,1,0,1,1,2,3,5,
%U A161553 1,6,0,6,6,5,4,2,6,1,0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,8,4,3,7,1,8,0,8,8
%N A161553 Table which contains in row n the fundamental Pisano period of the Fibonacci sequence (mod n).
%C A161553 The length of the n-th row (the length of the period) is A001175(n).
%H A161553 Alois P. Heinz, <a href="/A161553/b161553.txt">Rows n = 1..200, flattened</a>
%H A161553 J. D. Fulton and W. L. Morris, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf">On arithmetical functions related to the Fibonacci numbers</a>, Acta Arithm. 16 (1969) 106-110.
%H A161553 Wayne Peng, <a href="http://arxiv.org/abs/1511.05645">ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields</a>, arXiv:1511.05645 [math.NT], 2015.
%H A161553 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisanoPeriod.html">Pisano Period</a>.
%H A161553 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pisano_period">Pisano period</a>.
%e A161553 F(n) mod 1 {0},
%e A161553 F(n) mod 2 {0,1,1},
%e A161553 F(n) mod 3 {0,1,1,2,0,2,2,1},
%e A161553 F(n) mod 4 {0,1,1,2,3,1},
%e A161553 F(n) mod 5 {0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1},
%e A161553 F(n) mod 6 {0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2,5,1},
%e A161553 F(n) mod 7 {0,1,1,2,3,5,1,6,0,6,6,5,4,2,6,1},
%e A161553 F(n) mod 8 {0,1,1,2,3,5,0,5,5,2,7,1},
%e A161553 F(n) mod 9 {0,1,1,2,3,5,8,4,3,7,1,8,0,8,8,7,6,4,1,5,6,2,8,1},
%e A161553 F(n) mod 10 {0,1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8, 1,9,0,9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3,2,5,7,2,9,1}.
%t A161553 per[1] = 1; per[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k + 1], n] == 1, Return[k]]];
%t A161553 row[n_] := Table[Mod[Fibonacci[k], n], {k, 0, per[n]-1}];
%t A161553 Array[row, 9] // Flatten (* _Jean-François Alcover_, Oct 30 2018 *)
%o A161553 (PARI) row(n)={my(L=List([0]), X=Mod([1,1;1,0],n), I=Mod([1,0;0,1],n), M=X); while(M<>I, M*=X; listput(L, lift(M[2,2]))); Vec(L)} \\ _Andrew Howroyd_, Mar 05 2023
%Y A161553 Cf. A000045, A001175.
%Y A161553 Main diagonal gives A002708.
%Y A161553 Row sums give A214300.
%K A161553 nonn,look,tabf
%O A161553 1,8
%A A161553 _Alexander Adamchuk_, Jun 13 2009
%E A161553 Moved into the keyword:tabf category by _R. J. Mathar_, Oct 04 2009