A129345 a(2n) = A001542(n+1), a(2n+1) = A038761(n+1); a Pellian-related sequence.

2, 9, 12, 53, 70, 309, 408, 1801, 2378, 10497, 13860, 61181, 80782, 356589, 470832, 2078353, 2744210, 12113529, 15994428, 70602821, 93222358, 411503397, 543339720, 2398417561, 3166815962, 13979001969, 18457556052, 81475594253, 107578520350, 474874563549

Offset: 0

Author

Creighton Dement, Apr 10 2007

Keywords

• easy
• nonn

Comments

Summation of -a(n) and A129346 returns twice Pell numbers A000129 (apart from signs; starting from 2nd term of A000129).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A129346, A000129, A001542, A038761.

Programs

Mathematica

CoefficientList[Series[(2 + 9 x - x^3)/((x^2 + 2 x - 1) (x^2 - 2 x - 1)), {x, 0, 29}], x] (* Michael De Vlieger, May 26 2016 *)
LinearRecurrence[{0,6,0,-1},{2,9,12,53},30] (* Harvey P. Dale, Apr 29 2025 *)

PARI

Vec((2+9*x-x^3)/((x^2+2*x-1)*(x^2-2*x-1)) + O(x^40)) \\ Colin Barker, May 26 2016

Formula

G.f.: (2+9*x-x^3)/((x^2+2*x-1)*(x^2-2*x-1)).

From Colin Barker, May 26 2016: (Start)

a(n) = ((-1-sqrt(2))^(1+n)-(-1+sqrt(2))^(1+n)+(1-sqrt(2))^n*(-4+3*sqrt(2))+(1+sqrt(2))^n*(4+3*sqrt(2)))/(2*sqrt(2)).

a(n) = 6*a(n-2)-a(n-4) for n>3.

(End)