A114647 Expansion of (3 -4*x -3*x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

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

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