A106293 Period of the Lucas 3-step sequence A001644 mod n.

1, 1, 13, 4, 31, 13, 48, 8, 39, 31, 10, 52, 168, 48, 403, 16, 96, 39, 360, 124, 624, 10, 553, 104, 155, 168, 117, 48, 140, 403, 331, 32, 130, 96, 1488, 156, 469, 360, 2184, 248, 560, 624, 308, 20, 1209, 553, 46, 208, 336, 155, 1248, 168, 52, 117, 310, 48, 4680, 140

Offset: 1

Author

T. D. Noe, May 02 2005

Keywords

• nonn
• changed

Comments

This sequence can differ from the corresponding Fibonacci sequence (A046738) only when n is a multiple of 2 or 11 because the discriminant of the characteristic polynomial x^3-x^2-x-1 is -44. [Clarified by Avery Diep, Aug 22 2025]

a(n) divides A046738(n). - Avery Diep, Aug 22 2025

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Crossrefs

Cf. A046738 (period of Fibonacci 3-step sequence mod n), A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1).

Programs

Mathematica

n=3; Table[p=i; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 60}]

Formula

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).