# [seqfan] Experiment: period of a(n) mod p (after ditching rational expressions and linear recurrences)

Georgi Guninski guninski at guninski.com
Wed Jan 26 15:24:23 CET 2011

```Did an experiment for determining the period mod p of a(n) (after ditching rational expressions and linear recurrences).

116 sequences popped up from a subset of oeis.

The main motivation was this could probably help compute some sequences - compute them mod enough small primes then use the Chinese remainder theorem.

The web interface is at:
http://stefan.guninski.com/cgi/oeis-modp1.cgi?q=%25N&full=0

Some relations:

\$r\$ is the period. relation f(p,r)=0 is given.

A066377 Number of numbers m <= n such that floor(sqrt(m)) divides m.
3*p = r
(^^^^ this seems strange to me. found doubling for it too.)

A000111 Euler or up/down numbers: expansion of sec x + tan x . Also number of alternating permutations on n letters.
-1/2*p^3 + 9/4*p^2*r - 5/2*p*r^2 + 3/4*r^3 - 9/4*p*r + 7/4*r^2 + 3/2*p - 1

A000255 a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
r = 2*p

A001053 a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.
(2*p - r) * (6*p^2 + 3*p*r + r^2)

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of Young tableaux with n cells
p = r

```