OFFSET
0,3
COMMENTS
A divisibility sequence by construction: a(n) divides a(m) if n divides m.
LINKS
Jianing Song, Table of n, a(n) for n = 0..1000
Wikipedia, Elliptic curves over finite fields
Index entries for linear recurrences with constant coefficients, signature (4,2,20,-25).
FORMULA
a(n) = (5^n + 1 - (-1+2*i)^n - (-1-2*i)^n)/8, i = sqrt(-1).
G.f.: (1/(1-5*x) + 1/(1-x) - (2+2*x)/(1+2*x+5*x^2))/8.
E.g.f.: (exp(5*x) + exp(x) - 2*exp(-x)*cos(2*x))/8.
EXAMPLE
For n = 1, the points on y^2 = x^3 - x in GF(5) are the point of infinity, (-1,0), (0,0), (1,0), (2,+-1), and (-2,+-2), so a(1) = 8/8 = 1. This implies that the number of points in GF(5^n) is 5^n + 1 - alpha^n - beta^n, where alpha + beta = 5+1-8 = -2 and alpha*beta = 5.
MATHEMATICA
LinearRecurrence[{4, 2, 20, -25}, {0, 1, 4, 13}, 30] (* Paolo Xausa, Jan 22 2026 *)
PROG
(PARI) a(n) = (5^n + 1 - (-1+2*I)^n - (-1-2*I)^n)/8
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Mar 17 2025
STATUS
approved