A190969 a(n) = 5*a(n-1) - 8*a(n-2), with a(0)=0, a(1)=1.

0, 1, 5, 17, 45, 89, 85, -287, -2115, -8279, -24475, -56143, -84915, 24569, 802165, 3814273, 12654045, 32756041, 62547845, 50690897, -246928275, -1640168551, -6225416555, -18005734367, -40225339395, -57080822039, 36398604965, 638639601137, 2902009165965

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

• sign
• easy

Comments

Let S(p):=Sum_{k=0..p-1} a(4k)*binomial(2k,k)^3/(-4096)^k. Zhi-Wei Sun conjectured that S(p) == 0 (mod p^2) for every odd prime p, and also S(p) == 0 (mod p^3) for any odd prime p == 1,2,4 (mod 7). - Zhi-Wei Sun, Mar 13 2013

(a(n) + ((-1)^n)*n) mod 7 = 0 for n > 0; division yields following signed integer sequence: {0, 1, 2, 7, 12, 13, -42, -301, -1184, -3495, -8022, -12129, 3508, 114597, ...} with g.f.: (x - x^2)/((1 + x)^2 * (1 - 5*x + 8*x^2)). - Alexander R. Povolotsky, Mar 13 2013

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..100
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
• Index entries for linear recurrences with constant coefficients, signature (5,-8).

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences).

Programs

Mathematica

LinearRecurrence[{5,-8}, {0,1}, 50]

Maxima

a[0]:0$ a[1]:1$ a[n]:=5*a[n-1] - 8*a[n-2]$ makelist(a[n], n,0, 50); /* Martin Ettl, Oct 21 2012 */

Formula

G.f.: x/(1-5x+8*x^2). - Philippe Deléham, Oct 12 2011