A366968 -id:A366968 - cp's OEIS frontend

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

0, 0, 1, 5, 15, 36, 71, 131, 216, 346, 511, 756, 1042, 1441, 1907, 2527, 3207, 4128, 5097, 6371, 7737, 9442, 11213, 13538, 15848, 18734, 21744, 25423, 29077, 33743, 38238, 43818, 49440, 56104, 62694, 70979, 78749, 88154, 97580, 108790, 119450, 132680, 145021, 159974

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363607.
Cf. A366968, A366969, A366970.
Cf. A024916, A064602, A064603, A366967.

a(n) = sum(k=3, n, binomial(k, 3)*(n\k));

from math import isqrt, comb
def A366971(n): return -comb((s:=isqrt(n))+1,4)*(s+1)+sum(comb((q:=n//w)+1,4)+(q+1)*comb(w,3) for w in range(1,s+1)) # Chai Wah Wu, Oct 30 2023

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

a(n) = (A064603(n) - 3*A064602(n) + 2*A024916(n))/6. - Chai Wah Wu, Oct 30 2023

A134867 A010766 * A000012.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 8, 4, 2, 1, 10, 5, 3, 2, 1, 14, 8, 5, 3, 2, 1, 16, 9, 6, 4, 3, 2, 1, 20, 12, 8, 6, 4, 3, 2, 1, 23, 14, 10, 7, 5, 4, 3, 2, 1, 27, 17, 12, 9, 7, 5, 4, 3, 2, 1, 29, 18, 13, 10, 8, 6, 5, 4, 3, 2, 1, 35, 23, 17, 13, 10, 8, 6, 5, 4, 3, 2, 1, 37, 24, 18, 14, 11, 9, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

			First few rows of the triangle:
   1;
   3,  1;
   5,  2,  1;
   8,  4,  2, 1;
  10,  5,  3, 2, 1;
  14,  8,  5, 3, 2, 1;
  16,  9,  6, 4, 3, 2, 1;
  20, 12,  8, 6, 4, 3, 2, 1;
  23, 14, 10, 7, 5, 4, 3, 2, 1;
  27, 17, 12, 9, 7, 5, 4, 3, 2, 1;
  ...

Column k=1..4 give: A006218, A002541, A366968, A366972.
Row sums give A024916.
Cf. A010766, A135539.

t = Table[Sum[Floor[n/h], {h, k, n}], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A134867 array *)
TableForm[t]    (* A134867 sequence *)
(* Clark Kimberling, Oct 11 2014 *)

T(n, k) = sum(j=k, n, n\j); \\ Seiichi Manyama, Oct 30 2023

A010766 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial row sums of A010766 starting fromt the right.
G.f. of column k: 1/(1-x) * Sum_{j>=1} x^(k*j)/(1-x^j) = 1/(1-x) * Sum_{j>=k} x^j/(1-x^j). - Seiichi Manyama, Oct 30 2023

More terms from Seiichi Manyama, Oct 30 2023

A366969 a(n) = Sum_{k=3..n} (k-2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 16, 24, 32, 43, 52, 69, 80, 97, 114, 136, 151, 179, 196, 227, 252, 281, 302, 347, 373, 408, 441, 486, 513, 570, 599, 651, 692, 739, 780, 854, 889, 942, 991, 1066, 1105, 1186, 1227, 1300, 1367, 1432, 1477, 1582, 1634, 1716, 1781, 1868, 1919, 2024

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A152771.

Cf. A006218, A024916, A366968, A366970, A366971.

PARI
```
a(n) = sum(k=3, n, (k-2)*(n\k));
```

from math import isqrt
def A366969(n): return n+(-(s:=isqrt(n))*(s*(s-2)-7)+sum(((q:=n//w)+1)*(q+(w<<1)-8) for w in range(1,s+1))>>1) # Chai Wah Wu, Oct 30 2023

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

a(n) = n + A024916(n) - 2*A006218(n). - Chai Wah Wu, Oct 30 2023

A366970 a(n) = Sum_{k=3..n} binomial(k-1,2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 36, 60, 89, 131, 176, 245, 311, 404, 502, 631, 751, 926, 1079, 1295, 1501, 1756, 1987, 2330, 2612, 2978, 3332, 3779, 4157, 4707, 5142, 5736, 6278, 6926, 7508, 8336, 8966, 9785, 10555, 11533, 12313, 13427, 14288, 15449, 16521, 17742, 18777, 20306

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363610.

Cf. A006218, A024916, A064602, A366968, A366969, A366971.

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

from math import isqrt
def A366970(n): return (-(s:=isqrt(n))*(s*(s**2-(s<<1)-1)+8)+sum(((q:=n//w)+1)*(q*(q-4)+3*(w**2-3*w+4)) for w in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

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

a(n) = (A064602(n)-3*A024916(n))/2 + A006218(n). - Chai Wah Wu, Oct 30 2023

cp's OEIS Frontend

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A134867 A010766 * A000012.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A366969 a(n) = Sum_{k=3..n} (k-2) * floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A366970 a(n) = Sum_{k=3..n} binomial(k-1,2) * floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula