A366723 -id:A366723 - cp's OEIS frontend

A366395 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+2,3).

1, 3, 10, 16, 32, 49, 78, 100, 152, 194, 261, 318, 410, 489, 631, 717, 871, 1014, 1205, 1351, 1617, 1806, 2083, 2300, 2641, 2903, 3333, 3612, 4048, 4450, 4947, 5289, 5923, 6367, 7041, 7548, 8252, 8805, 9683, 10245, 11107, 11873, 12820, 13497, 14719, 15526, 16655

Offset: 1

Seiichi Manyama, Oct 24 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums of A365007.
Cf. A078471, A366659, A366723.
Cf. A364970.

Array[Sum[(-1)^(k - 1)*Binomial[Floor[#/k] + 2, 3], {k, #}] &, 56] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(n\k+2, 3));

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

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).

A366659 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 4, 15, 30, 66, 115, 200, 295, 471, 659, 946, 1259, 1715, 2194, 2920, 3591, 4561, 5585, 6916, 8216, 10082, 11823, 14124, 16389, 19350, 22174, 26004, 29435, 33931, 38445, 43902, 48925, 55767, 61941, 69831, 77275, 86415, 94968, 106094, 115874, 128216, 140214, 154405

Offset: 1

Seiichi Manyama, Oct 24 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums of A366813.

Cf. A078471, A366395, A366723.

Array[Sum[(-1)^(k - 1)*Binomial[Floor[#/k] + 3, 4], {k, #}] &, 56] (* Michael De Vlieger, Oct 25 2023 *)

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

a(n) = Sum_{k=1..n} binomial(k+2,3) * (floor(n/k) mod 2).

G.f.: -1/(1-x) * Sum_{k>=1} (-x)^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1+x^k).

A366814 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+3,4).

Original entry on oeis.org

1, 4, 16, 29, 71, 115, 211, 289, 511, 649, 1002, 1253, 1821, 2174, 3146, 3505, 4846, 5605, 7316, 8099, 10852, 11653, 14951, 16333, 20546, 21935, 27916, 28904, 35961, 38620, 46377, 48113, 59922, 61204, 74096, 77024, 91391, 93959, 113766, 114059, 135752, 140654, 163186

Offset: 1

Seiichi Manyama, Oct 24 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums give A366723.

Cf. A000593, A365007, A366813.

Table[DivisorSum[n, (-1)^(n/# - 1)*Binomial[# + 3, 4] &], {n, 56}] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d-1)*binomial(d+3, 4));

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

A366395 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+2,3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366659 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+3,4).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366814 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+3,4).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula