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

Author

Creighton Dement, Feb 18 2006

Keywords

• easy
• nonn

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).

Crossrefs

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

Programs

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

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 *)

PARI

Vec((-1-3*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, May 26 2016

Sage

[(10*lucas_number1(n+1,2,-1) -3*(1+(-1)^n))/4 for n in (0..30)] # G. C. Greubel, May 24 2021

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