A387628 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k+1,2*k+1).
1, 3, 5, 9, 29, 81, 185, 429, 1093, 2785, 6817, 16613, 41181, 102441, 253049, 623693, 1541557, 3814929, 9430545, 23297397, 57577997, 142345721, 351858985, 869614109, 2149341925, 5312698977, 13131636417, 32457015109, 80223121469, 198288112969, 490110342873
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,-1,4,4,0,-4).
Programs
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Magma
[&+[2^k* Binomial(2*n-4*k+1, 2*k+1): k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
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Mathematica
Table[Sum[2^k*Binomial[2*n-4*k+1,2*k+1],{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *) -
PARI
a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k+1, 2*k+1));
Formula
G.f.: (1+x-2*x^3)/((1+x-2*x^3)^2 - 4*x).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3) + 4*a(n-4) - 4*a(n-6).