A024324 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 = A000201 (lower Wythoff sequence).

0, 0, 3, 4, 6, 8, 9, 11, 20, 23, 27, 29, 33, 37, 39, 43, 60, 65, 70, 74, 80, 84, 89, 94, 98, 104, 131, 137, 143, 150, 157, 163, 169, 176, 183, 189, 195, 202, 241, 248, 256, 265, 272, 281, 289, 296, 306, 313, 321, 329, 337, 346, 397, 406, 416, 425, 436, 445, 454, 466, 474, 484

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, A024325, A024326, A024327.

Cf. A000201, A023531.

Programs

Magma

b:= func< n,j | IsIntegral((Sqrt(8*j+9) -3)/2) select Fibonacci(n-j+1) else 0 >;
A024324:= func< n | (&+[b(n,j): j in [1..Floor((n+1)/2)]]) >;
[A024324(n) : n in [1..80]]; // G. C. Greubel, Jan 28 2022

Mathematica

Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += Floor[n*GoldenRatio^2]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Jan 28 2022 *)

PARI

my(phi=quadgen(5)); a(n) = my(L=n>>1,m=2,ret=0); n--; while(n>L, ret += floor(n*phi); n-=(m++)); ret; \\ Kevin Ryde, Feb 03 2022

Sage

def b(n,j): return floor( (n+1-j)*(1+sqrt(5))/2 ) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
def A024324(n): return sum( b(n,k) for k in (1..((n+1)//2)) )
[A024324(n) for n in (1..80)] # G. C. Greubel, Jan 28 2022

Formula

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A000201(n-j+1).

Extensions

a(62) corrected by Sean A. Irvine, Jun 27 2019