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A114696 Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence. 4
1, 6, 15, 40, 97, 238, 575, 1392, 3361, 8118, 19599, 47320, 114241, 275806, 665855, 1607520, 3880897, 9369318, 22619535, 54608392, 131836321, 318281038, 768398399, 1855077840, 4478554081, 10812186006, 26102926095, 63018038200, 152139002497, 367296043198 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.

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

a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Harvey P. Dale, Jan 23 2014

a(n) = (-3 - (-1)^n + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2)))/2. - Colin Barker, May 26 2016

From G. C. Greubel, May 24 2021: (Start)

a(n) = 3*A000129(n+1) + A000129(n) - (3 + (-1)^n)/2.

a(n) = (1/2)*(A002203(n+2) - 3 - (-1)^n). (End)

MAPLE

Q:= proc(n) option remember; # Q=A002203

    if n<2 then 2

  else 2*Q(n-1) + Q(n-2)

    fi; end:

seq((Q(n+2) -3 -(-1)^n)/2, n=0..40); # G. C. Greubel, May 24 2021

MATHEMATICA

CoefficientList[Series[(1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, 2, -2, -1}, {1, 6, 15, 40}, 30] (* Harvey P. Dale, Jan 23 2014 *)

PROG

(PARI) Vec((1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^30)) \\ Colin Barker, May 26 2016

(Sage) [(lucas_number2(n+2, 2, -1) -3 -(-1)^n)/2 for n in (0..30)] # G. C. Greubel, May 24 2021

CROSSREFS

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

Cf. A114647, A114688, A114689, A114695, A114697.

Sequence in context: A273748 A272847 A273829 * A106368 A100491 A220030

Adjacent sequences:  A114693 A114694 A114695 * A114697 A114698 A114699

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Feb 18 2006

STATUS

approved

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Last modified September 26 20:57 EDT 2022. Contains 357050 sequences. (Running on oeis4.)