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A114688 Expansion of (1 +3*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 5 14:56 EDT 2022. Contains 357259 sequences. (Running on oeis4.)