A094723 a(n) = Pell(n+2) - 2^n.

1, 3, 8, 21, 54, 137, 344, 857, 2122, 5229, 12836, 31413, 76686, 186833, 454448, 1103921, 2678674, 6494037, 15732284, 38089677, 92173782, 222961529, 539145416, 1303349513, 3150038746, 7611815613, 18390447188, 44426264421, 107310084894

Offset: 0

Author

Paul Barry, May 23 2004

Keywords

• easy
• nonn

Comments

Binomial transform of A052955.

The sequence b(n) = 2*a(n), n >= -1, is an elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 23 and 464, lead to the b(n) sequence. For the central square these vectors lead to the companion sequence A175658. - Johannes W. Meijer, Aug 15 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2).

Crossrefs

Cf. A000129.

Programs

Magma

I:=[1, 3, 8]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2012

Mathematica

LinearRecurrence[{4,-3, -2},{1,3,8},40] (* Vincenzo Librandi, Jun 24 2012 *)

Formula

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

a(n) = ((1+sqrt(2))^n*(3*sqrt(2)/4+1) - (3*sqrt(2)/4-1)*(1-sqrt(2))^n) - 2^n.

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3). - Vincenzo Librandi, Jun 24 2012