A330466 -id:A330466 - cp's OEIS frontend

A285914 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 4, 1, 2, 0, 0, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 4, 1, 2, 3, 0, 5, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 3, 4, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 4, 0, 0, 1, 2, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 2, 0, 0, 5, 0, 1, 0, 0, 4, 0, 0, 1, 2, 3, 0, 0, 6

Offset: 1

Omar E. Pol, Apr 28 2017

Conjecture 1: T(n,k) is the number of parts in the partition of n into k consecutive parts, if T(n,k) > 0.
Conjecture 2: row sums give A204217, which should be also the total number of parts in all partitions of n into consecutive parts.
(The conjectures are true. See Joerg Arndt's proof in the Links section.) - Omar E. Pol, Jun 14 2017
From Omar E. Pol, May 05 2020: (Start)
Theorem: Let T(n,k) be an irregular triangle read by rows in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th (m+2)-gonal number, with n >= 1, k >= 1, m >= 0. T(n,k) is also the number of parts in the partition of n into k consecutive parts that differ by m, including n as a valid partition. Hence the sum of row n gives the total number of parts in all partitions of n into consecutive parts that differ by m.
About the above theorem, this is the case for m = 1. For m = 0 see the triangle A127093, in which row sums give A000203. For m = 2 see the triangle A330466, in which row sums give A066839 (conjectured). For m = 3 see the triangle A330888, in which row sums give A330889.
Note that there are infinitely many triangles of this kind, with m >= 0. Also, every triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve. (End)

			Triangle begins (rows 1..28):
1;
1;
1,  2;
1,  0;
1,  2;
1,  0,  3;
1,  2,  0;
1,  0,  0;
1,  2,  3;
1,  0,  0,  4;
1,  2,  0,  0;
1,  0,  3,  0;
1,  2,  0,  0;
1,  0,  0,  4;
1,  2,  3,  0,  5;
1,  0,  0,  0,  0;
1,  2,  0,  0,  0;
1,  0,  3,  4,  0;
1,  2,  0,  0,  0;
1,  0,  0,  0,  5;
1,  2,  3,  0,  0,  6;
1,  0,  0,  4,  0,  0;
1,  2,  0,  0,  0,  0;
1,  0,  3,  0,  0,  0;
1,  2,  0,  0,  5,  0;
1,  0,  0,  4,  0,  0;
1,  2,  3,  0,  0,  6;
1,  0,  0,  0,  0,  0,  7;
...
In accordance with the conjectures, for n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. These partitions are formed by 1, 2, 3 and 5 consecutive parts respectively, so the 15th row of the triangle is [1, 2, 3, 0, 5].
Illustration of initial terms:
Row                                                         _
1                                                         _|1|
2                                                       _|1 _|
3                                                     _|1  |2|
4                                                   _|1   _|0|
5                                                 _|1    |2 _|
6                                               _|1     _|0|3|
7                                             _|1      |2  |0|
8                                           _|1       _|0 _|0|
9                                         _|1        |2  |3 _|
10                                      _|1         _|0  |0|4|
11                                    _|1          |2   _|0|0|
12                                  _|1           _|0  |3  |0|
13                                _|1            |2    |0 _|0|
14                              _|1             _|0   _|0|4 _|
15                            _|1              |2    |3  |0|5|
16                          _|1               _|0    |0  |0|0|
17                        _|1                |2     _|0 _|0|0|
18                      _|1                 _|0    |3  |4  |0|
19                    _|1                  |2      |0  |0 _|0|
20                  _|1                   _|0     _|0  |0|5 _|
21                _|1                    |2      |3   _|0|0|6|
22              _|1                     _|0      |0  |4  |0|0|
23            _|1                      |2       _|0  |0  |0|0|
24          _|1                       _|0      |3    |0 _|0|0|
25        _|1                        |2        |0   _|0|5  |0|
26      _|1                         _|0       _|0  |4  |0 _|0|
27    _|1                          |2        |3    |0  |0|6 _|
28   |1                            |0        |0    |0  |0|0|7|
...
Note that the k's are placed exactly below the k-th horizontal line segment of every row.
The above structure is related to the triangle A237591, also to the left-hand part of the triangle A237593, and also to the left-hand part of the front view of the pyramid described in A245092.

Joerg Arndt, Proof of the conjectures of A204217 and A285914, SeqFan Mailing Lists, Jun 03 2017.

Row n has length A003056(n).
Column k starts in row A000217(k).
The number of positive terms in row n is A001227(n), the number of partitions of n into consecutive parts.
Cf. A000203, A066839, A196020, A204217, A235791, A236104, A237048, A237591, A237593, A245092, A261699, A285898, A262626, A330889.
Triangles of the same family are A127093, this sequence, A330466, A330888.

With[{nn = 6}, Table[Boole[If[EvenQ@ k, Mod[(n - k/2), k] == 0, Mod[n, k] == 0]] k, {n, nn (nn + 3)/2}, {k, Floor[((Sqrt[8 n + 1] - 1)/2)]}]] // Flatten (* Michael De Vlieger, Jun 15 2017, after Python by Indranil Ghosh *)

t(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0); \\ A237048
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(k*t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

from sympy import sqrt
import math
def a237048(n, k):
    return int(n%k == 0) if k%2 else int(((n - k//2)%k) == 0)
def T(n, k): return k*a237048(n, k)
for n in range(1, 29): print([T(n, k) for k in range(1, int(math.floor((sqrt(8*n + 1) - 1)/2)) + 1)]) # Indranil Ghosh, Apr 30 2017

T(n,k) = k*A237048(n,k).

A330888 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 3, n >= 1, k >= 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, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 4, 1, 2, 0, 0, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 4, 1, 2, 3, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 3, 4, 1, 2, 0, 0, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 4, 1, 2, 0, 0, 5

Offset: 1

Omar E. Pol, Apr 30 2020

Since the trivial partition n is counted, so T(n,1) = 1.

This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th pentagonal number.

			Triangle begins (rows 1..26):
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0;
1, 2, 0;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0, 4;
1, 2, 0, 0;
1, 0, 3, 0;
1, 2, 0, 0;
1, 0, 0, 4;
...
For n = 21 there are three partitions of 21 into consecutive parts that differ by 3, including 21 as a partition. They are [21], [12, 9] and [10, 7, 4]. The number of parts of these partitions are 1, 2 and 3 respectively, so the 21st row of the triangle is [1, 2, 3].

Cf. A000326, A330887.

Other triangles of the same family are A127093, A285914, A330466.

A330888 := proc(n,k)
    k*A330887(n,k) ;
end proc:
for n from 1 to 40 do
    for k from 1 do
        if n>= A000325(k) then
            printf("%d,",A330888(n,k)) ;
        else
            break;
        end if;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Oct 02 2020

T(n,k) = k*A330887(n,k).

A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 3, 1, 1, 6, 1, 1, 1, 1, 12, 3, 3, 1, 1, 1, 8, 4, 1, 1, 1, 1, 1, 15, 3, 3, 3, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 18, 6, 3, 3, 3, 1, 1, 1, 1, 1, 12, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 14, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 3, 6, 3, 3, 3, 3, 1

Offset: 1

Omar E. Pol, May 01 2020

The one-part partition n = n is included in the count.
The column k is related to (k+2)-gonal numbers, assuming that 2-gonals are the nonnegative numbers, 3-gonals are the triangular numbers, 4-gonals are the squares, 5-gonals are the pentagonal numbers, and so on.
Note that the number of parts for T(n,0) = A000203(n), equaling the sum of the divisors of n.
For fixed k>0, Sum_{j=1..n} T(j,k) ~ 2^(3/2) * n^(3/2) / (3*sqrt(k)). - Vaclav Kotesovec, Oct 23 2024

			Square array starts:
   n\k|   0  1  2  3  4  5  6  7  8  9 10 11 12
   ---+---------------------------------------------
   1  |   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   2  |   3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   3  |   4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   4  |   7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   5  |   6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   6  |  12, 4, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, ...
   7  |   8, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, ...
   8  |  15, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, ...
   9  |  13, 6, 4, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, ...
  10  |  18, 5. 3. 1. 3. 1, 3, 1, 3, 1, 1, 1, 1, ...
  11  |  12, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, ...
  12  |  28, 4, 6, 4, 3, 1, 3, 1, 3, 1, 3, 1, 1, ...
  ...
For n = 9 we have that:
For k = 0 the partitions of 9 into consecutive parts that differ by 0 (or simply: the partitions of 9 into equal parts) are [9], [3,3,3], [1,1,1,1,1,1,1,1,1]. In total there are 13 parts, so T(9,0) = 13.
For k = 1 the partitions of 9 into consecutive parts that differ by 1 (or simply: the partitions of 9 into consecutive parts) are [9], [5,4], [4,3,2]. In total there are six parts, so T(9,1) = 6.
For k = 2 the partitions of 9 into consecutive parts that differ by 2 are [9], [5, 3, 1]. In total there are four parts, so T(9,2) = 4.

Columns k: A000203 (k=0), A204217 (k=1), A066839 (k=2), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6), A377300 (k=7), A377301 (k=8).
Triangles whose row sums give the column k: A127093 (k=0), A285914 (k=1), A330466 (k=2) (conjectured), A330888 (k=3), A334462 (k=4), A334540 (k=5), A339947 (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).
Tables of partitions related to column k: A010766 (k=0), A286001 (k=1), A332266 (k=2), A334945 (k=3), A334618 (k=4).
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. A051735, A237048, A303300, A330887, A334460, A334465, A334946.
Cf. A038040, A245579, A060872, A334467, A327262, A334733, A334953.
Cf. A057145, A323345.

nmax = 14;
col[k_] := col[k] = CoefficientList[Sum[n x^(n(k n - k + 2)/2)/(1 - x^n), {n, 1, nmax}] + O[x]^(nmax+1), x];
T[n_, k_] := col[k][[n+1]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

The g.f. for column k is Sum_{n>=1} n*x^(n*(k*n-k+2)/2)/(1-x^n). (For proof, see A330889. - N. J. A. Sloane, Nov 21 2020)

A334462 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 4, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 0, 1, 2, 0, 4, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 2, 3, 4, 1, 0, 0, 0, 1, 2, 0

Offset: 1

Omar E. Pol, May 05 2020

Since the trivial partition n is counted, so T(n,1) = 1.
This is also an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th hexagonal number.
This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.
For a general theorem about the triangles of this family see A285914.

			Triangle begins (rows 1..28):
1;
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0;
1, 2, 0;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0;
1, 2, 0;
1, 0, 3;
1, 2, 0, 4;
...
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 number of parts of these partitions are 1, 2, 4 respectively, so the 28th row of the triangle is [1, 2, 0, 4].

Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), this sequence (d=4), A334540 (d=5).

Cf. A000384, A334460, A334461.

T(n,k) = k*A334460(n,k).

A334540 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 5, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th heptagonal number (A000566).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 4, 1, 2, 0, 0, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 4, 1, 2, 3, 0, 1, 0, 0, 0, 1

Offset: 1

Omar E. Pol, May 05 2020

Since the trivial partition n is counted, so T(n,1) = 1.
This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th heptagonal number.
This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.
For a general theorem about the triangles of this family see A285914.

			Triangle begins (rows 1..27):
1;
1;
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0;
1, 2, 0;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
...
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 number of parts of these partitions are 1, 2, 3 respectively, so the 27th row of the triangle is [1, 2, 3].

Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), A334462 (d=4), this sequence (d=5).

Cf. A000566, A334465.

A334540 := proc(n,k)
        k*A334465(n,k) ;
end proc: # R. J. Mathar, Oct 02 2020

T(n,k) = k*A334465(n,k).

A332266 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 2, where the m-th block starts with m, m >= 1, and the first element of column k is in row k^2.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 08 2020

This triangle can be interpreted as a table of partitions into consecutive parts that differ by 2 (see the Example section).

			Triangle begins:
   1;
   2;
   3;
   4,  1;
   5,  3;
   6,  2;
   7,  4;
   8,  3;
   9,  5,  1;
  10,  4,  3;
  11,  6,  5;
  12,  5,  2;
  13,  7,  4;
  14,  6,  6;
  15,  8,  3;
  16,  7,  5,  1;
  17,  9,  7,  3;
  18,  8,  4,  5;
  19, 10,  6,  7;
  20,  9,  8,  2;
  21, 11,  5,  4;
  22, 10,  7,  6;
  23, 12,  9,  8;
  24, 11,  6,  3;
  25, 13,  8,  5,  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 2:
.   ---------------------------------------------------------
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],[1];|  4,  1;|  4,  1; |  4,  1;|
5   |     |     |     |  5, [3];| [5], 3;|  5,  3; |  5,  3;|
6   |     |     |     |         |        | [6],[2];|  6,  2;|
7   |     |     |     |         |        |  7, [4];| [7], 4;|
.   ---------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts that differ by 2 (but with the parts in increasing order) are [6] and [2, 4]. 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 6.
.
Figures H..L show the location (in the columns of the table) of the partitions of 8..12 (respectively) into consecutive parts that differ by 2:
.    -----------------------------------------------------------------------
Fig:        H            I            J              K             L
.    -----------------------------------------------------------------------
. n:        8            9            10             11            12
Row  -----------------------------------------------------------------------
1    |  1;        |  1;         |   1;        |   1;        |   1;         |
1    |  2;        |  2;         |   2;        |   2;        |   2;         |
3    |  3;        |  3;         |   3;        |   3;        |   3;         |
4    |  4,  1;    |  4,  1;     |   4,  1;    |   4,  1;    |   4,  1;     |
5    |  5,  3;    |  5,  3;     |   5,  3;    |   5,  3;    |   5,  3;     |
6    |  6,  2;    |  6,  2;     |   6,  2;    |   6,  2;    |   6,  2;     |
7    |  7,  4;    |  7,  4;     |   7,  4;    |   7,  4;    |   7,  4;     |
8    | [8],[3];   |  8,  3;     |   8,  3;    |   8,  3;    |   8,  3;     |
9    |  9, [5], 1;| [9], 5, [1];|   9,  5,  1;|   9,  5,  1;|   9,  5,  1; |
10   |            | 10,  4, [3];| [10],[4], 3;|  10,  4,  3;|  10,  4;  3; |
11   |            | 11,  6, [5];|  11, [6], 5;| [11], 6,  5,|  11,  6;  5; |
12   |            |             |             |             | [12],[5],[2];|
13   |            |             |             |             |  13, [7],[4];|
14   |            |             |             |             |  14,  6, [6];|
.    -----------------------------------------------------------------------
Figure I: for n = 9 the partitions of 9 into consecutive parts that differ by 2(but with the parts in increasing order) are [9] and [1, 3, 5]. These partitions have one part and three parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 9.
.
Illustration of initial terms arranged into a triangular structure:
.                                                           _
.                                                         _|1|
.                                                       _|2  |
.                                                     _|3   _|
.                                                   _|4    |1|
.                                                 _|5     _|3|
.                                               _|6      |2  |
.                                             _|7       _|4  |
.                                           _|8        |3   _|
.                                         _|9         _|5  |1|
.                                       _|10         |4    |3|
.                                     _|11          _|6   _|5|
.                                   _|12           |5    |2  |
.                                 _|13            _|7    |4  |
.                               _|14             |6     _|6  |
.                             _|15              _|8    |3   _|
.                           _|16               |7      |5  |1|
.                         _|17                _|9     _|7  |3|
.                       _|18                 |8      |4    |5|
.                     _|19                  _|10     |6   _|7|
.                   _|20                   |9       _|8  |2  |
.                 _|21                    _|11     |5    |4  |
.               _|22                     |10       |7    |6  |
.             _|23                      _|12      _|9   _|8  |
.           _|24                       |11       |6    |3   _|
.          |25                         |13       |8    |5  |1|
...
The number of horizontal line segments in the n-th row of the diagram equals A038548(n), the number of partitions of n into consecutive parts that differ by 2.

Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), this sequence (d=2), A334945 (d=3), A334618(d=4).

Cf. A038548, A060872, A066839, A303300, A330466.

A334947 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 6, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th octagonal number (A000567).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 0

Offset: 1

Omar E. Pol, May 27 2020

Since the trivial partition n is counted, so T(n,1) = 1.
This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th octagonal number.
This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.
For a general theorem about the triangles of this family see A285914.

			Triangle begins (rows 1..24).
1;
1;
1;
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
...
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]. There are 1, 2 and 3 parts respectively, so the 24th row of this triangle is [1, 2, 3].

Row sums give A334949.
Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), A334462 (d=4), A334540 (d=5), this sequence (d=6).
Cf. A000567, A334946, A334948, A334953.

T(n,k) = k*A334946(n,k).

cp's OEIS Frontend

A285914 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

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A330888 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 3, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th pentagonal number (A000326).

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A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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A334462 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 4, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

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A334540 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 5, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th heptagonal number (A000566).

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A332266 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 2, where the m-th block starts with m, m >= 1, and the first element of column k is in row k^2.

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A334947 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 6, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th octagonal number (A000567).

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