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

0,2

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).

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

Mathematica

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

Prog

(PARI) Vec((-1-4*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^30)) \\ Colin Barker, May 26 2016
(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
(Sage) [3*lucas_number1(n+1, 2, -1) -(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021

Crossrefs

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

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

Sequence in context: A301356 A037243 A107710 * A280532 A342564 A338267

Adjacent sequences: A114686 A114687 A114688 * A114690 A114691 A114692

Keyword

easy,nonn

Author

Creighton Dement, Feb 18 2006

Status

approved