A114696 Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

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

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