A214300 -id:A214300 - cp's OEIS frontend

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

Alexander Adamchuk, Jun 13 2009

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

			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}.

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.

Cf. A000045, A001175.
Main diagonal gives A002708.
Row sums give A214300.

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 *)

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

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

A088551 Fibonacci winding number: the number of 'mod n' operations in one cycle of the Fibonacci sequence modulo n.

Original entry on oeis.org

1, 3, 2, 8, 11, 7, 4, 11, 28, 3, 9, 12, 23, 19, 9, 16, 11, 7, 28, 5, 12, 23, 9, 48, 40, 35, 19, 4, 59, 12, 19, 15, 16, 39, 9, 36, 6, 27, 28, 19, 19, 43, 11, 59, 23, 15, 9, 55, 148, 35, 38, 52, 35, 6, 21, 31, 16, 26, 57, 28, 12, 21, 43, 68, 51, 67, 14, 19, 119, 32, 7, 72, 112, 99, 5, 33

Offset: 2

R C Johnson (bob.johnson(AT)dur.ac.uk), Nov 19 2003

If pi(n) is the n-th Pisano number (A001175) then a(n) is usually about pi(n)/2 - and in any case a(n) > pi(n)/4.

			a(8)=4 because one cycle of the Fibonacci numbers modulo 8 is 0, 1, 1, 2, 3, 5; 0, 5, 5; 2, 7; 1; - including 4 'mod 8' operations, each marked with a semi-colon.

T. D. Noe, Table of n, a(n) for n=2..10000
R. C. Johnson, Fibonacci Numbers and Resources.
M. Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Int. Seq. 14 (2011) # 11.9.1.

Cf. A001175, A015134, A214300.

(* pp = Pisano period = A001175 *) pp[1] = 1;
pp[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k + 1], n] == 1, Return[k]]];
a[n_] := Sum[Mod[Fibonacci[k], n], {k, 1, pp[n]}]/n;
Table[a[n], {n, 2, 77}] (* Jean-François Alcover, Sep 05 2017 *)

n*a(n) = sum{k=1..A001175(n)} fibonacci(k) mod n. [Mircea Merca, Jan 03 2011]

More terms from T. D. Noe

Edited by Ray Chandler, Oct 26 2006

A376083 Row sums of A376081.

Original entry on oeis.org

0, 1, 7, 5, 40, 22, 47, 18, 93, 100, 45, 132, 180, 110, 275, 84, 304, 210, 134, 580, 131, 78, 527, 240, 1200, 388, 927, 652, 160, 590, 373, 328, 587, 644, 1355, 444, 1404, 286, 1075, 1140, 821, 278, 1847, 542, 2625, 1102, 767, 504, 2681, 2500, 1815, 2204, 2860

Offset: 1

Paolo Xausa, Sep 10 2024

nonn

See A376081 for more information.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A214300, A376081.

A376081row[n_] := If[n < 3, {n - 1}, Module[{k = 1}, NestWhileList[Mod[2 * Fibonacci[++k] - 1, n] &, 1, {#, #2} != {1, n-1} &, {3, 2}]]];
Array[Total[A376081row[#]] &, 100]

a(n) = Sum_{k = 0..A376082(n)-1} A376081(n,k).

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

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A088551 Fibonacci winding number: the number of 'mod n' operations in one cycle of the Fibonacci sequence modulo n.

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A376083 Row sums of A376081.

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