A024325 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 = A001950 (upper Wythoff sequence).

0, 0, 5, 7, 10, 13, 15, 18, 33, 38, 44, 48, 54, 60, 64, 70, 98, 106, 114, 121, 130, 137, 145, 153, 160, 169, 213, 223, 233, 244, 255, 265, 275, 286, 297, 307, 317, 328, 391, 403, 416, 430, 442, 456, 469, 481, 496, 508, 521, 534, 547, 561, 644, 659, 675, 690, 707, 722, 737, 755

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, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024326, A024327.

Cf. A001950, A023531.

Programs

Magma

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
A024325:= func< n | (&+[A023531(j)*Floor((n-j+1)*(3+Sqrt(5))/2): j in [1..Floor((n+1)/2)]]) >;
[A024325(n) : n in [1..80]]; // G. C. Greubel, Jan 28 2022

Mathematica

A023531[n_] := SquaresR[1, 8n+9]/2;
a[n_]:= a[n]= Sum[A023531[j]*Floor[(n-j+1)*GoldenRatio^2], {j,Floor[(n+1)/2]}];
Table[a[n], {n, 80}] (* G. C. Greubel, Jan 28 2022 *)

Sage

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
def A023325(n): return sum( A023531(j)*floor(((n-j+1)*(3+sqrt(5)))/2) for j in (1..((n+1)//2)) )
[A023325(n) for n in (1..80)] # G. C. Greubel, Jan 28 2022

Formula

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A001950(n-j+1).