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A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 26 22:02 EDT 2022. Contains 357051 sequences. (Running on oeis4.)