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

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, 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, 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

Offset

1,8

Comments

The length of the n-th row (the length of the period) is A001175(n).

Links

Alois P. Heinz, Rows n = 1..200, flattened

J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithm. 16 (1969) 106-110.

Wayne Peng, ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields, arXiv:1511.05645 [math.NT], 2015.

Eric Weisstein's World of Mathematics, Pisano Period.

Wikipedia, Pisano period.

Example

F(n) mod 1 {0},
F(n) mod 2 {0,1,1},
F(n) mod 3 {0,1,1,2,0,2,2,1},
F(n) mod 4 {0,1,1,2,3,1},
F(n) mod 5 {0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1},
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},
F(n) mod 7 {0,1,1,2,3,5,1,6,0,6,6,5,4,2,6,1},
F(n) mod 8 {0,1,1,2,3,5,0,5,5,2,7,1},
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},
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}.

Mathematica

per[1] = 1; per[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k + 1], n] == 1, Return[k]]];
row[n_] := Table[Mod[Fibonacci[k], n], {k, 0, per[n]-1}];
Array[row, 9] // Flatten (* Jean-François Alcover, Oct 30 2018 *)

Prog

(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

Crossrefs

Cf. A000045, A001175.

Main diagonal gives A002708.

Row sums give A214300.

Sequence in context: A155100 A076880 A082115 * A366444 A256232 A099751

Adjacent sequences: A161550 A161551 A161552 * A161554 A161555 A161556

Keyword

nonn,look,tabf,changed

Author

Alexander Adamchuk, Jun 13 2009

Extensions

Moved into the keyword:tabf category by R. J. Mathar, Oct 04 2009

Status

approved