OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).
FORMULA
G.f.: x*(3 + 6*x + 6*x^2 - 8*x^3 - 7*x^4 + x^5 - 4*x^6 + 2*x^7)/((1 - x - x^2)*(1 - x^2 - x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
From G. C. Greubel, Jan 17 2022: (Start)
a(2*n) = 6*F(2*n+3) + F(2*n+1) - (n+6)*F(n+3) - (n+1)*F(n+1).
a(2*n+1) = 6*F(2*n+2) + F(2*n) - (n+6)*F(n+2) - (n+1)*F(n), where F(n) = A000045(n). (End)
MATHEMATICA
a[n_]:= With[{F=Fibonacci}, 6*F[n+3] +F[n+1] - (1/2)*((1+(-1)^n)*(((n+2)/2 )*LucasL[(n+4)/2] + 5*F[(n+6)/2]) - (1-(-1)^n)*(((n+3)/2)*LucasL[(n+3)/2] +5*F[(n+5)/2] ))];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022
(Sage)
def A024314_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024314_list(41); a[1:] # G. C. Greubel, Jan 17 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved