A366937 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+1,2) * floor(n/k).

1, -1, 6, -6, 10, -7, 22, -26, 26, -16, 51, -54, 38, -41, 101, -83, 71, -72, 119, -143, 123, -66, 211, -230, 111, -151, 279, -216, 220, -182, 315, -397, 237, -207, 467, -430, 274, -279, 599, -519, 343, -423, 524, -665, 557, -250, 879, -874, 380, -612, 874, -776

Offset

1,3

Links

Table of n, a(n) for n=1..52.

Formula

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

Prog

(PARI) a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+1, 2)*(n\k));
(Python)
from math import isqrt
def A366937(n): return (((s:=isqrt(m:=n>>1))*(s+1)**2*((s<<2)+5)<<1)-(t:=isqrt(n))*(t+1)**2*(t+2)-sum((((q:=m//w)+1)*(q*((q<<2)+5)+6*w*((w<<1)+1))<<1) for w in range(1, s+1))+sum(((q:=n//w)+1)*(q*(q+2)+3*w*(w+1)) for w in range(1, t+1)))//6 # Chai Wah Wu, Oct 29 2023

Crossrefs

Partial sums of A320900.

Cf. A024919, A366938, A366939.

Cf. A364970, A366395.

Sequence in context: A301690 A365786 A365783 * A096474 A220439 A363324

Adjacent sequences: A366934 A366935 A366936 * A366938 A366939 A366940

Keyword

sign

Author

Seiichi Manyama, Oct 29 2023

Status

approved