A106288 Number of different orbit lengths of the 3-step recursion mod n.

1, 3, 2, 4, 2, 6, 3, 5, 3, 6, 4, 8, 3, 6, 4, 6, 3, 9, 3, 8, 6, 8, 2, 10, 3, 5, 4, 8, 3, 12, 2, 7, 8, 5, 6, 12, 2, 6, 6, 10, 3, 12, 3, 11, 6, 6, 3, 12, 5, 9, 6, 7, 3, 12, 8, 9, 6, 6, 2, 16, 3, 6, 7, 8, 6, 16, 2, 6, 4, 12, 2, 15, 3, 6, 6, 8, 10, 10, 3, 12, 5, 5, 3, 16, 6, 7, 6, 14, 2, 18, 6, 8, 4, 6, 6

Offset

1,2

Comments

Consider the 3-step recursion x(k)=x(k-1)+x(k-2)+x(k-3) mod n. For any of the n^3 initial conditions x(1), x(2) and x(3) in Zn, the recursion has a finite period. Each of these n^3 vectors belongs to exactly one orbit. In general, there are only a few different orbit lengths for each n. For n=8, there are 5 different lengths: 1, 2, 4, 8 and 16. The maximum possible length of an orbit is A046738(n), the period of the Fibonacci 3-step sequence mod n.

Links

Table of n, a(n) for n=1..95.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Crossrefs

Cf. A106285 (orbits of 3-step sequences), A106307 (primes that yield a simple orbit structure in 3-step recursions).

Sequence in context: A304182 A256284 A322979 * A013633 A334662 A016559

Adjacent sequences: A106285 A106286 A106287 * A106289 A106290 A106291

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved