A131717 -id:A131717 - cp's OEIS frontend

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]

A131017 Period 6: repeat [1, 1, 2, -1, 2, 1].

Original entry on oeis.org

1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 2

Offset: 0

Paul Curtz, Sep 23 2007

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A131717, A131718.

Flatten[Table[{1, 1, 2, -1, 2, 1}, {20}]] (* Wesley Ivan Hurt, Jun 24 2022 *)
PadRight[{},120,{1,1,2,-1,2,1}] (* Harvey P. Dale, Oct 10 2023 *)

|a(n)| = A131718(n).
a(n) = A131717(n+1) - A131717(n).
O.g.f.: (1+x+2*x^2-x^3+2*x^4+x^5)/((1-x)*(1+x)*(x^2+x+1)*(x^2-x+1)). - R. J. Mathar, Jul 07 2008
a(n) = (3+2*(-1)^n+cos(n*Pi/3)-3*cos(n*Pi/3)^2+3*sin(n*Pi/3)^2)/3. - Wesley Ivan Hurt, Jun 24 2022

Edited by R. J. Mathar, Jul 07 2008

cp's OEIS Frontend

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

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