A024327 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 = A014306.

0, 0, 1, 1, 0, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 3, 4, 4, 3, 4, 4, 3, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 7, 8, 7, 8, 7

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, A024325, A024326, A024328.

Cf. A014306, A023531.

Programs

Mathematica

A014306:= With[{ms= Table[m(m+1)(m+2)/6, {m,0,20}]}, Table[If[MemberQ[ms, n], 0, 1], {n,0,150}]];
Table[t=0; m=3; p=BitShiftRight[n]; n--; While[n>p, t += A014306[[n+1]]; n -= m++]; t, {n, 120}] (* G. C. Greubel, Feb 17 2022 *)

Sage

nmax=120
@CachedFunction
def b_list(N):
    A = []
    for m in range(ceil((6*N)^(1/3))):
        A.extend([0]*(binomial(m+2, 3) - len(A)) + [1])
    return A
A023533 = b_list(nmax+5)
def A014306(n): return 1 - A023533[n]
def b(n, j): return A014306(n-j+1) if ((sqrt(8*j+9) -3)/2).is_integer() else 0
@CachedFunction
def A024327(n): return sum( b(n, j) for j in (1..floor((n+1)/2)) )
[A024327(n) for n in (1..nmax)] # G. C. Greubel, Feb 17 2022

Formula

a(n) = Sum_{k=1..floor((n+1)/2)} A023531(k)*A014306(n-k+1). - G. C. Greubel, Feb 17 2022

Extensions

Title corrected by Sean A. Irvine, Jun 30 2019