A129744 a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).

2, 4, 14, 32, 82, 196, 478, 1152, 2786, 6724, 16238, 39200, 94642, 228484, 551614, 1331712, 3215042, 7761796, 18738638, 45239072, 109216786, 263672644, 636562078, 1536796800, 3710155682, 8957108164, 21624372014, 52205852192

Offset: 1

Author

N. J. A. Sloane, May 13 2007

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Crossrefs

Cf. A002203, A000129, A001350, A113224.

Programs

Maple

u:=1+sqrt(2): v:=1-sqrt(2): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n),n=1..33); # Emeric Deutsch, May 13 2007

Mathematica

Table[Simplify[ -((1 + Sqrt[2])^n - 1)*((1 - Sqrt[2])^n - 1)], {n, 1, 30}] (* Stefan Steinerberger, May 15 2007 *)

PARI

w = quadgen(8); vector(30, n, -((1+w)^n-1)*((1-w)^n-1)) \\ Michel Marcus, Mar 21 2015

PARI

Vec(2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x))+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015

Formula

a(2n) = A002203(2n)-2. a(2n+1) = A002203(2n+1). - R. J. Mathar, corrected Dec 05 2007.

G.f.: 2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x)).

From Peter Bala, Mar 19 2015: (Start)

a(n) = -det(I - M^n) where I is the 2X2 identity matrix and M = [2, 1; 1, 0]. Cf. A001350.

a(n) = 2*A113224(n-1).

This is divisibility sequence, that is, if n | m then a(n) | a(m).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n > 4. - Seiichi Manyama, Jun 07 2018

Extensions

More terms from Emeric Deutsch and Stefan Steinerberger, May 13 2007