A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

0, 1, 4, 11, 21, 40, 61, 96, 135, 191, 246, 337, 415, 528, 646, 801, 937, 1145, 1316, 1568, 1802, 2089, 2342, 2737, 3047, 3451, 3841, 4338, 4744, 5358, 5823, 6474, 7060, 7758, 8384, 9294, 9960, 10835, 11657, 12717, 13537, 14739, 15642, 16881, 18025, 19314, 20395

Offset: 1

Author

Seiichi Manyama, Oct 30 2023

Keywords

• nonn

Crossrefs

Partial sums of A069153.

Cf. A002541, A236632.

Cf. A024916, A064602, A366971.

Programs

PARI

a(n) = sum(k=2, n, binomial(k, 2)*(n\k));

Python

from math import isqrt
def A366967(n): return ((s:=isqrt(n))**2*(1-s**2)+sum((q:=n//k)*(3*k*(k-1)+q**2-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

Formula

G.f.: 1/(1-x) * Sum_{k>=1} x^(2*k)/(1-x^k)^3 = 1/(1-x) * Sum_{k>=2} binomial(k,2) * x^k/(1-x^k).

a(n) = (A064602(n)-A024916(n))/2. - Chai Wah Wu, Oct 30 2023