A114689 Expansion of (1 +4*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

1, 6, 13, 36, 85, 210, 505, 1224, 2953, 7134, 17221, 41580, 100381, 242346, 585073, 1412496, 3410065, 8232630, 19875325, 47983284, 115841893, 279667074, 675176041, 1630019160, 3935214361, 9500447886, 22936110133, 55372668156, 133681446445, 322735561050

Offset: 0

Author

Creighton Dement, Feb 18 2006

Keywords

• easy
• nonn

Comments

Elements of odd index give match to A075848: 2*n^2 + 9 is a square. Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Crossrefs

Cf. A000129, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114695, A114696, A114697.

Programs

Magma

I:=[1,6,13,36]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 24 2021

Mathematica

Table[3*Fibonacci[n+1, 2] -1-(-1)^n, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)

PARI

Vec((-1-4*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^30)) \\ Colin Barker, May 26 2016

Sage

[3*lucas_number1(n+1,2,-1) -(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021

Formula

G.f.: (1 +4*x -x^2)/((1-x)*(1+x)*(1-2*x-x^2)).

From Colin Barker, May 26 2016: (Start)

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

a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) for n>3.

(End)

a(n) = 3*A000129(n+1) - (1 + (-1)^n). - G. C. Greubel, May 24 2021