A135532 a(n) = 2*a(n-1) + a(n-2), with a(0)= -1, a(1)= 3.

-1, 3, 5, 13, 31, 75, 181, 437, 1055, 2547, 6149, 14845, 35839, 86523, 208885, 504293, 1217471, 2939235, 7095941, 17131117, 41358175, 99847467, 241053109, 581953685, 1404960479, 3391874643, 8188709765, 19769294173, 47727298111, 115223890395, 278175078901, 671574048197

Offset

0,2

Comments

Double binomial transform of [1, 3, -5, 13, -31, 75, -181, ...] = the Pell-like sequence A048655: (1, 5, 11, 27, 65, 157, ...). - Gary W. Adamson, Jul 23 2008

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1).

Formula

From R. J. Mathar, Feb 23 2008: (Start)

O.g.f.: (-1 + 5*x)/(1 - 2*x - x^2).

a(n) = 5*A000129(n) - A000129(n+1). (End)

a(n) = (1/2)*( (2*sqrt(2) - 1)*(1 + sqrt(2))^n - (1 + 2*sqrt(2))*(1 - sqrt(2))^n ), with n>=0. - Paolo P. Lava, Jun 09 2008

a(n) = ((3+sqrt(2))*(1+sqrt(2))^n + (3-sqrt(2))*(1-sqrt(2))^n)/2 with offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

a(n) = (1/2)*(A002203(n) + 2*A002203(n-1)). - G. C. Greubel, May 22 2021

Mathematica

LinearRecurrence[{2, 1}, {-1, 3}, 25] (* G. C. Greubel, Oct 17 2016 *)

Prog

(PARI) a(n)=([0, 1; 1, 2]^n*[-1; 3])[1, 1] \\ Charles R Greathouse IV, Oct 17 2016
(Magma) I:=[-1, 3]; [n le 2 select I[n] else 2*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, May 22 2021
(Sage) [(lucas_number2(n, 2, -1) + 2*lucas_number2(n-1, 2, -1))/2 for n in (0..30)] # G. C. Greubel, May 22 2021

Crossrefs

Cf. A000129, A002203, A048655.

Sequence in context: A354167 A350392 A281874 * A127600 A262237 A051401

Adjacent sequences: A135529 A135530 A135531 * A135533 A135534 A135535

Keyword

sign,easy

Author

Paul Curtz, Feb 21 2008

Extensions

More terms from R. J. Mathar, Feb 23 2008

Status

approved