A114688 Expansion of (1 +3*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

1, 5, 11, 30, 71, 175, 421, 1020, 2461, 5945, 14351, 34650, 83651, 201955, 487561, 1177080, 2841721, 6860525, 16562771, 39986070, 96534911, 233055895, 562646701, 1358349300, 3279345301, 7917039905, 19113425111, 46143890130, 111401205371, 268946300875

Offset

0,2

Comments

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 +3*x -x^2)/((1-x)*(1+x)*(1-2*x-x^2)).

a(0)=1, a(1)=5, a(2)=11, a(3)=30, a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4). - Harvey P. Dale, Dec 18 2012

a(n) = (-6 - 6*(-1)^n + 5*sqrt(2)*( (1+sqrt(2))^(1+n) - (1-sqrt(2))^(1+n) ))/8. - Colin Barker, May 26 2016

a(n) = (10*A000129(n+1) - 3*(1 + (-1)^n))/4. - G. C. Greubel, May 24 2021

Maple

Pell:= proc(n) option remember;
    if n<2 then n
  else 2*Pell(n-1) + Pell(n-2)
    fi; end:
seq((10*Pell(n+1) -3*(1+(-1)^n))/4, n=0..40); # G. C. Greubel, May 24 2021

Mathematica

CoefficientList[Series[(-1-3x+x^2)/((1-x)(x+1)(x^2+2x-1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 2, -2, -1}, {1, 5, 11, 30}, 40] (* Harvey P. Dale, Dec 18 2012 *)

Prog

(PARI) Vec((-1-3*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, May 26 2016
(Magma) I:=[1, 5, 11, 30]; [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) [(10*lucas_number1(n+1, 2, -1) -3*(1+(-1)^n))/4 for n in (0..30)] # G. C. Greubel, May 24 2021

Crossrefs

Cf. A000129, A100828, A114647, A114689, A114695, A114696, A114697.

Sequence in context: A209659 A266820 A335455 * A257717 A192194 A239842

Adjacent sequences: A114685 A114686 A114687 * A114689 A114690 A114691

Keyword

easy,nonn

Author

Creighton Dement, Feb 18 2006

Status

approved