A366938 -id:A366938 - cp's OEIS frontend

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

Seiichi Manyama, Oct 29 2023

sign

Partial sums of A320900.
Cf. A024919, A366938, A366939.
Cf. A364970, A366395.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+1, 2)*(n\k));

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

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).

A366939 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+3,4) * floor(n/k).

Original entry on oeis.org

1, -3, 13, -26, 45, -70, 141, -228, 283, -366, 636, -879, 942, -1232, 1914, -2331, 2515, -3090, 4226, -5313, 5539, -6114, 8837, -10558, 9988, -11947, 15969, -17705, 18256, -20364, 26013, -30592, 29330, -31874, 42222, -47034, 44357, -49602, 64164, -69115, 66637, -74017

Offset: 1

Seiichi Manyama, Oct 29 2023

sign

Cf. A024919, A366937, A366938.

Cf. A365439, A366723.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+3, 4)*(n\k));

from math import isqrt
from sympy import rf
def A366939(n): return ((rf(s:=isqrt(m:=n>>1),3)*(s+1)*((s**2<<2)+13*s+8)<<3)-rf(t:=isqrt(n),5)*(t+1)+sum((((q:=m//w)+1)*(-q*(q+2)*((q**2<<2)+13*q+8)-5*w*(w+1)*((r:=w<<1)+1)*(r+3))<<3) for w in range(1,s+1))+sum(rf(q:=n//w,5)+5*(q+1)*rf(w,4) for w in range(1,t+1)))//120 # Chai Wah Wu, Oct 29 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^5 = -1/(1-x) * Sum_{k>=1} binomial(k+3,4) * (-x)^k/(1-x^k).

cp's OEIS Frontend

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

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