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A114647 Expansion of (3 -4*x -3*x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence. 5
3, 2, 7, 12, 31, 70, 171, 408, 987, 2378, 5743, 13860, 33463, 80782, 195027, 470832, 1136691, 2744210, 6625111, 15994428, 38613967, 93222358, 225058683, 543339720, 1311738123, 3166815962, 7645370047, 18457556052, 44560482151, 107578520350, 259717522851 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.: (3 -4*x -3*x^2)/((1-x)*(1+x)*(1-2*x-x^2)).

a(n) = A000129(n+1) + 2*A059841(n). - R. J. Mathar, Nov 10 2009

From Colin Barker, May 26 2016: (Start)

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

MATHEMATICA

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

PROG

(PARI) Vec((3-4*x-3*x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, May 26 2016

(Magma) I:=[3, 2, 7, 12]; [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) [lucas_number1(n+1, 2, -1) +(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021

CROSSREFS

Cf. A000129, A059841, A100828, A114688, A114689, A114695, A114696, A114697.

Sequence in context: A143332 A255919 A212189 * A234750 A052546 A260016

Adjacent sequences:  A114644 A114645 A114646 * A114648 A114649 A114650

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Feb 18 2006

STATUS

approved

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Last modified October 7 08:31 EDT 2022. Contains 357270 sequences. (Running on oeis4.)