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

Author

Paul Curtz, Feb 21 2008

Keywords

• sign
• easy

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).

Crossrefs

Cf. A000129, A002203, A048655.

Programs

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

Mathematica

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

PARI

a(n)=([0,1; 1,2]^n*[-1;3])[1,1] \\ Charles R Greathouse IV, Oct 17 2016

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

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) = ((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

Extensions

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