A334463 -id:A334463 - cp's OEIS frontend

A327262 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 4.

1, 2, 3, 4, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 30, 32, 17, 54, 19, 40, 42, 44, 23, 72, 25, 52, 54, 84, 29, 90, 31, 96, 66, 68, 35, 144, 37, 76, 78, 120, 41, 126, 43, 132, 135, 92, 47, 192, 49, 150, 102, 156, 53, 162, 110, 168, 114, 116, 59, 300, 61, 124, 126, 192, 130, 264, 67, 204, 138, 210

Offset: 1

Omar E. Pol, Apr 30 2020

nonn

The one-part partition n = n is included in the count.

			For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. The sum of the parts is [28] + [16 + 12] + [13 + 9 + 5 + 1] = 84, so a(28) = 84.

Cf. A334460, A334461.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), this sequence (k=4), A334733 (k=5).

pn4[n_]:=Total[Flatten[Select[IntegerPartitions[n],Union[Abs[Differences[#]]]=={4}&]]]+n; Array[pn4,70] (* Harvey P. Dale, Nov 26 2023 *)

a(n) = n*A334461(n).

A334953 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 42, 44, 23, 72, 25, 52, 54, 56, 29, 90, 31, 64, 66, 68, 35, 108, 37, 76, 78, 120, 41, 126, 43, 132, 90, 92, 47, 192, 49, 100, 102, 156, 53, 162, 55, 168, 114, 116, 59, 240, 61, 124, 126, 192, 130, 198, 67, 204, 138, 210

Offset: 1

Omar E. Pol, May 27 2020

nonn

The one-part partition n = n is included in the count.

			For n = 24 there are three partitions of 24 into consecutive parts that differ by 6, including 24 as a valid partition. They are [24], [15, 9] and [14, 8, 2]. The sum of all parts is [24] + [15 + 9] + [14 + 8 + 2] = 72, so a(24) = 72.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), this sequence (k=6).

Cf. A334946, A334947, A334948, A334949.

a(n) = n*A334948(n).

A334733 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 36, 38, 20, 63, 22, 46, 48, 50, 26, 81, 28, 58, 60, 62, 32, 99, 68, 70, 72, 74, 76, 117, 40, 82, 126, 86, 44, 135, 92, 94, 96, 98, 100, 153, 52, 106, 162, 165, 56, 171, 116, 118, 180, 122, 124, 189, 64, 195, 198, 134, 68, 207, 210

Offset: 1

Omar E. Pol, May 09 2020

nonn

The one-part partition n = n is included in the count.

			For n = 27 there are three partitions of 27 into consecutive parts that differ by 5, including 27 as a valid partition. They are [27], [16, 11] and [14, 9, 4]. The sum of all parts is [27] + [16 + 11] + [14 + 9 + 4] = 81, so a(27) = 81.

Sequences of the same family where the parts differs by k are: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), this sequence (k=5).

Cf. A334465, A334541.

a(n) = n*A334541(n).

A334945 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 3, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th pentagonal number (A000326).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 6, 4, 7, 2, 8, 5, 9, 3, 10, 6, 11, 4, 12, 7, 1, 13, 5, 4, 14, 8, 7, 15, 6, 2, 16, 9, 5, 17, 7, 8, 18, 10, 3, 19, 8, 6, 20, 11, 9, 21, 9, 4, 22, 12, 7, 1, 23, 10, 10, 4, 24, 13, 5, 7, 25, 11, 8, 10, 26, 14, 11, 2, 27, 12, 6, 5, 28, 15, 9, 8, 29, 13, 12, 11, 30, 16, 7, 3

Offset: 1

Omar E. Pol, May 27 2020

This triangle can be interpreted as a table of partitions into consecutive parts that differ by 3 (see the Example section).

Also, every triangle of this family has the property that starting from row n the sum of k positive and consecutive terms in the column k is equal to n. - Omar E. Pol, Dec 18 2020

			Triangle begins:
   1;
   2;
   3;
   4;
   5,  1;
   6,  4;
   7,  2;
   8,  5;
   9,  3;
  10,  6;
  11,  4;
  12,  7,  1;
  13,  5,  4;
  14,  8,  7;
  15,  6,  2;
  16,  9,  5;
  17,  7,  8;
  18, 10,  3;
  19,  8,  6;
  20, 11,  9;
  21,  9,  4;
  22, 12,  7,  1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts that differ by 3:
.   -----------------------------------------------------
Fig:   A     B     C     D       E        F        G
.   -----------------------------------------------------
. n:   1     2     3     4       5        6        7
Row -----------------------------------------------------
1   | [1];|  1; |  1; |  1; |  1;     |  1;   |  1;     |
2   |     | [2];|  2; |  2; |  2;     |  2;   |  2;     |
3   |     |     | [3];|  3; |  3;     |  3;   |  3;     |
4   |     |     |     | [4];|  4;     |  4;   |  4;     |
5   |     |     |     |     | [5],[1];|  5, 1;|  5,  1; |
6   |     |     |     |     |  6, [4];| [6],4;|  6,  4; |
7   |     |     |     |     |         |       | [7],[2];|
8   |     |     |     |     |         |       |  8, [5];|
.   -----------------------------------------------------
Figure G: for n = 7 the partitions of 7 into consecutive parts that differ by 3 (but with the parts in increasing order) are [7] and [2, 5]. These partitions have one part and two parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 7.
.
Illustration of initial terms arranged into a triangular structure:
.                                                           _
.                                                         _|1|
.                                                       _|2  |
.                                                     _|3    |
.                                                   _|4     _|
.                                                 _|5      |1|
.                                               _|6       _|4|
.                                             _|7        |2  |
.                                           _|8         _|5  |
.                                         _|9          |3    |
.                                       _|10          _|6    |
.                                     _|11           |4     _|
.                                   _|12            _|7    |1|
.                                 _|13             |5      |4|
.                               _|14              _|8     _|7|
.                             _|15               |6      |2  |
.                           _|16                _|9      |5  |
.                         _|17                 |7       _|8  |
.                       _|18                  _|10     |3    |
.                     _|19                   |8        |6    |
.                   _|20                    _|11      _|9    |
.                 _|21                     |9        |4     _|
.                |22                       |12       |7    |1|
...
The number of horizontal line segments in the n-th row of the diagram equals A117277(n), the number of partitions of n into consecutive parts that differ by 3.

Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), A332266 (d=2), this sequence (d=3), A334618(d=4).

Cf. A000326, A117277, A330887, A330888, A330889, A334463.

A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

Original entry on oeis.org

1, 4, 1, 6, 2, 1, 12, 6, 2, 1, 10, 4, 3, 2, 1, 24, 10, 8, 3, 2, 1, 14, 12, 5, 4, 3, 2, 1, 32, 14, 12, 10, 4, 3, 2, 1, 27, 8, 7, 6, 5, 4, 3, 2, 1, 40, 27, 16, 14, 12, 5, 4, 3, 2, 1, 22, 20, 18, 8, 7, 6, 5, 4, 3, 2, 1, 72, 22, 20, 18, 16, 14, 6, 5, 4, 3, 2, 1, 26, 24, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, May 05 2020

			Array begins:
     k  0   1   2   3   4   5   6   7   8   9  10
   n +------------------------------------------------
   1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2 |  4,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
   3 |  6,  6,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
   4 | 12,  4,  8,  4,  4,  4,  4,  4,  4,  4,  4, ...
   5 | 10, 10,  5, 10,  5,  5,  5,  5,  5,  5,  5, ...
   6 | 24, 12, 12,  6, 12,  6,  6,  6,  6,  6,  6, ...
   7 | 14, 14,  7, 14,  7, 14,  7,  7,  7,  7,  7, ...
   8 | 32,  8, 16,  8, 16,  8, 16,  8,  8,  8,  8, ...
   9 | 27, 27, 18, 18,  9, 18,  9, 18,  9,  9,  9, ...
  10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...
...

Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).
Every diagonal starting with 1 gives A000027.
Sequences of number of parts related to column k: A000203 (k=0), A204217 (k=1), A066839 (k=2) (conjectured), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6).
Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).
Polygonal  numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).
Cf. A245579, A323345, A334466.

nmax = 13;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
T[n_, k_] := n col[k][[n]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

T(n,k) = n*A323345(n,k).

cp's OEIS Frontend

A327262 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 4.

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A334953 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 6.

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A334733 a(n) is the sum of all parts of all partitions of n into consecutive parts that differ by 5.

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A334945 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 3, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th pentagonal number (A000326).

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A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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