A023613 Convolution of Fibonacci numbers and A023533.

1, 1, 2, 4, 6, 10, 16, 26, 42, 69, 111, 180, 291, 471, 762, 1233, 1995, 3228, 5223, 8452, 13675, 22127, 35802, 57929, 93731, 151660, 245391, 397051, 642442, 1039493, 1681935, 2721428, 4403363, 7124791

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..4000

Crossrefs

Cf. A000045, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[Fibonacci(k)*A023533(n+2-k): k in [1..n+1]]): n in [0..50]]; // G. C. Greubel, Jul 14 2022

Mathematica

Join[{1,1}, Table[Sum[Fibonacci[m+2 -Binomial[j+3,3]], {j,0,n}], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* G. C. Greubel, Jul 14 2022 *)

Sage

#Assuming A023533 is available as an array
for n in range(34):
    print(n, sum([A023533[k]*fibonacci(n+2-k) for k in range(1,n+2)]))
# Danny Rorabaugh, Mar 14 2015

Formula

a(n) = Sum_{k=1..n+1} A000045(k)*A023533(n+2-k). - Danny Rorabaugh, Mar 13 2015