OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,4,-1).
FORMULA
G.f.: (1-2*x-x^2)/((1-2*x-x^2)^2 - 8*x^3).
a(n) = 4*a(n-1) - 2*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A387694(n-1) + (-1)^floor(n/2).
MATHEMATICA
CoefficientList[Series[(1-2*x-x^2)/((1-2*x-x^2)^2 - 8*x^3), {x, 0, 26}], x] (* Stefano Spezia, Sep 06 2025 *)
Table[Sum[2^(n-2*k)*Binomial[2*n-2k, 2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 06 2025 *)
PROG
(PARI) a(n) = sum(k=0, n\2, 2^(n-2*k) * binomial(2*n-2*k, 2*k));
(Magma) [&+[2^(n-2*k)* Binomial(2*n-2*k, 2*k): k in [0..Floor (n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 06 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Sep 06 2025
STATUS
approved