A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

1, 3, 9, 22, 55, 133, 323, 780, 1885, 4551, 10989, 26530, 64051, 154633, 373319, 901272, 2175865, 5253003, 12681873, 30616750, 73915375, 178447501, 430810379, 1040068260, 2510946901, 6061962063, 14634871029, 35331704122, 85298279275, 205928262673

Offset

0,2

Comments

Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').

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

a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).

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

From Raphie Frank, Oct 01 2012: (Start)

a(2*n) = A216134(2*n+1).

a(2*n+1) = A006452(2*n+3)-1.

Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)

a(n) = (2*A078343(n+2) - A010694(n))/4. - R. J. Mathar, Oct 02 2012

From Colin Barker, May 26 2016: (Start)

a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)

a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - G. C. Greubel, May 24 2021

Mathematica

Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)

Prog

(PARI) Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 24 2015
(Sage) [(4*lucas_number1(n+2, 2, -1) -2*lucas_number1(n+1, 2, -1) -3 +(-1)^n)/4 for n in (0..30)] # G. C. Greubel, May 24 2021

Crossrefs

Cf. A000129, A002203, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114689, A114695, A116699.

Sequence in context: A054442 A354648 A192663 * A032284 A329163 A296582

Adjacent sequences: A114694 A114695 A114696 * A114698 A114699 A114700

Keyword

easy,nonn

Author

Creighton Dement, Feb 18 2006

Status

approved