A024318 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (Fibonacci numbers).

0, 0, 1, 2, 3, 5, 8, 13, 26, 42, 68, 110, 178, 288, 466, 754, 1254, 2029, 3283, 5312, 8595, 13907, 22502, 36409, 58911, 95320, 154608, 250161, 404769, 654930, 1059699, 1714629, 2774328, 4488957, 7263285

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000045, A023531.

Programs

Magma

b:= func< n,j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;
A024318:= func< n | (&+[b(n,j): j in [1..Floor((n+1)/2)]]) >;
[A024318(n) : n in [1..80]]; // G. C. Greubel, Jan 19 2022

Mathematica

Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Fibonacci[n+1]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Jan 19 2022 *)

Sage

def b(n,j): return fibonacci(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
def A024318(n): return sum( b(n,j) for j in (1..floor((n+1)/2)) )
[A024318(n) for n in (1..120)] # G. C. Greubel, Jan 19 2022

Formula

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*Fibonacci(n-j+1). - G. C. Greubel, Jan 19 2022