OFFSET
0,2
COMMENTS
Corresponding y-values (A123116) are y(n) = c(n)*(-1 + d(n)), with c(n) and d(n) defined in formula section.
The pair (x,y) = (A001542(n), a(n)) satisfies the equation 2*x^4 + 2*x*y - y^2 = 0. - Alexander Samokrutov, Sep 04 2015
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).
FORMULA
a(n) = c(n)*(1 + d(n)) with: c(0) = 0, c(1) = 2 and c(n) = 6*c(n-1) - c(n-2), d(0) = 1, d(1) = 3 and d(n) = 6*d(n-1) - d(n-2).
For n>=4, a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4). - Max Alekseyev, Nov 13 2009
G.f.: 8*x*(1 -13*x +x^2)/((1-34*x+x^2)*(1-6*x+x^2)). - Colin Barker, Oct 24 2012
MATHEMATICA
CoefficientList[Series[8*x*(1-13*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2015 *)
Table[(Fibonacci[4*n, 2] + 2*Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jul 20 2021 *)
PROG
(PARI) concat(0, Vec(8*x*(1-13*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)) + O(x^20))) \\ Michel Marcus, Sep 05 2015
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 8*x*(1 -13*x +x^2)/((1-34*x+x^2)*(1-6*x+x^2)) )); // G. C. Greubel, Jul 20 2021
(Sage) [(1/2)*(lucas_number1(4*n, 2, -1) + 2*lucas_number1(2*n, 2, -1)) for n in (0..30)] # G. C. Greubel, Jul 20 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Sep 26 2006
EXTENSIONS
More terms from Max Alekseyev, Nov 13 2009
a(16) from Vincenzo Librandi, Sep 04 2015
Edited by Michel Marcus, Sep 05 2015
STATUS
approved