cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A352269 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into 2*k-1 consecutive parts, 1 <= k <= A351846(n). T(n,k) = 0 if no such partition exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 0, 1, 0, 1, 3, 1, 0, 1, 0, 1, 3, 1, 0, 1, 0, 1, 3, 5, 1, 0, 0, 1, 0, 0, 1, 3, 0, 1, 0, 0, 1, 0, 5, 1, 3, 0, 1, 0, 0, 1, 0, 0, 1, 3, 0, 1, 0, 5, 1, 0, 0, 1, 3, 0, 1, 0, 0, 7, 1, 0, 0, 0, 1, 3, 5, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 1, 0, 0, 0, 1, 0, 5, 7, 1, 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Mar 18 2022

Keywords

Comments

Column k lists 2*k-1 interleaved with 2*k-2 zeros, and the first element of column k is in row A000384(k).
This triangle is formed from the odd-indexed columns of the triangle A285914.
Conjecture 1: row sums give A341309.
Conjecture 2: A347529, A351819, A347263 have the same indices of zero and nonzero terms as A351824 and this sequence.

Examples

			Triangle begins:
-----------------------
   n / k   1  2  3  4
-----------------------
   1 |     1;
   2 |     1;
   3 |     1;
   4 |     1;
   5 |     1;
   6 |     1, 3;
   7 |     1, 0;
   8 |     1, 0;
   9 |     1, 3;
  10 |     1, 0;
  11 |     1, 0;
  12 |     1, 3;
  13 |     1, 0;
  14 |     1, 0;
  15 |     1, 3, 5;
  16 |     1, 0, 0;
  17 |     1, 0, 0;
  18 |     1, 3, 0;
  19 |     1, 0, 0;
  20 |     1, 0, 5;
  21 |     1, 3, 0;
  22 |     1, 0, 0;
  23 |     1, 0, 0;
  24 |     1, 3, 0;
  25 |     1, 0, 5;
  26 |     1, 0, 0;
  27 |     1, 3, 0;
  28 |     1, 0, 0, 7;
  ...

Links

Crossrefs

Row lengths give A351846.
The number of nonzeros terms in row n equals A082647(n).
Cf. A000384, A285914, A299765, A341309, A347529, A351819, A347263, A351824, A352425.

Programs

  • Mathematica
    A352269[rowmax_]:=Table[If[Divisible[n,2k-1],2k-1,0],{n,rowmax},{k,Floor[(Sqrt[8n+1]+1)/4]}];A352269[50] (* Paolo Xausa, Apr 09 2023 *)

Formula

T(n,k) = (2*k-1)*A351824(n,k). [Corrected by Paolo Xausa, Apr 09 2023]
T(n,k) = (2*k-1)*[(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - Paolo Xausa, Apr 09 2023

A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Mar 15 2022

Keywords

Comments

Conjecture: the total number of parts in all partitions of n into an odd number of consecutive parts equals the sum of odd divisors of n that are <= A003056(n). In other words: row n has A341309(n) terms.
The first partition with 2*m - 1 parts appears in the row A000384(m), m >= 1.

Examples

			Triangle begins:
   [1];
   [2];
   [3],
   [4];
   [5];
   [6], [3, 2, 1];
   [7];
   [8];
   [9], [4, 3, 2];
  [10];
  [11];
  [12], [5, 4, 3];
  [13];
  [14];
  [15], [6, 5, 4], [5, 4, 3, 2, 1];
  [16];
  [17];
  [18], [7, 6, 5];
  [19];
  [20], [6, 5, 4, 3, 2];
  [21], [8, 7, 6];
  [22];
  [23];
  [24], [9, 8, 7];
  [25], [7, 6, 5, 4, 3];
  [26];
  [27], [10, 9, 8];
  [28], [7, 6, 5, 4, 3, 2, 1];
  ...
In the diagram below the m-th staircase walk starts at row A000384(m).
The number of horizontal line segments in the n-th row equals A082647(n), the number of partitions of n into an odd number of consecutive parts, so we can find such partitions as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [6, 5, 4]. [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
                                                           _
                                                         _|1|
                                                       _|2  |
                                                     _|3    |
                                                   _|4      |
                                                 _|5       _|
                                               _|6        |3|
                                             _|7          |2|
                                           _|8           _|1|
                                         _|9            |4  |
                                       _|10             |3  |
                                     _|11              _|2  |
                                   _|12               |5    |
                                 _|13                 |4    |
                               _|14                  _|3   _|
                             _|15                   |6    |5|
                           _|16                     |5    |4|
                         _|17                      _|4    |3|
                       _|18                       |7      |2|
                     _|19                         |6     _|1|
                   _|20                          _|5    |6  |
                 _|21                           |8      |5  |
               _|22                             |7      |4  |
             _|23                              _|6      |3  |
           _|24                               |9       _|2  |
         _|25                                 |8      |7    |
       _|26                                  _|7      |6    |
     _|27                                   |10       |5   _|
    |28                                     |9        |4  |7|
...
The diagram is infinite.
For more information about the diagram see A286000.

Crossrefs

Subsequence of A299765.
Row sums give A352257.
Column 1 gives A000027.
Records give A000027.
Row n contains A082647(n) of the mentioned partitions.
Cf. A000384, A003056, A067742, A204217, A237048, A237591, A237593, A240542, A245092, A285574, A285901, A286000, A286001, A320051, A320137, A320142, A341309, A351824.

A341310 Sum of odd divisors of n that are > A003056(n).

Original entry on oeis.org

0, 0, 3, 0, 5, 0, 7, 0, 9, 5, 11, 0, 13, 7, 15, 0, 17, 9, 19, 0, 28, 11, 23, 0, 25, 13, 36, 0, 29, 15, 31, 0, 44, 17, 35, 9, 37, 19, 52, 0, 41, 21, 43, 11, 60, 23, 47, 0, 49, 25, 68, 13, 53, 27, 66, 0, 76, 29, 59, 15, 61, 31, 84, 0, 78, 33, 67, 17, 92, 35, 71, 0, 73, 37, 115, 19
Offset: 1

Views

Author

N. J. A. Sloane, Feb 14 2021

Keywords

Crossrefs

Cf. A000593, A003056, A082647, A131576, A333807, A341309.

A352446 Total number of parts in all partitions of n into an even number of consecutive parts.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 2, 0, 2, 4, 2, 0, 2, 4, 2, 0, 8, 4, 2, 0, 2, 4, 8, 0, 2, 4, 2, 0, 8, 4, 2, 8, 2, 4, 8, 0, 2, 4, 2, 8, 8, 4, 2, 0, 2, 4, 8, 8, 2, 4, 12, 0, 8, 4, 2, 8, 2, 4, 8, 0, 12, 4, 2, 8, 8, 4, 2, 0, 2, 4, 18, 8, 2, 16, 2, 0, 8, 4, 2, 8, 12, 4, 8, 0, 2, 16, 2
Offset: 1

Views

Author

Omar E. Pol, Mar 16 2022

Keywords

Examples

			For n = 21  the partitions of 21 into an even number of consecutive parts are [11, 10] and [6, 5, 4, 3, 2, 1]. The total number of parts in these two partitions is equal to 2 + 6 = 8, so a(21) = 8.
On the other hand consider the diagram below which is formed by the even-indexed staircase walks from the diagram of A286000.
The diagram is infinite and we have that:
The m-th staircase walk starts at row A014105(m).
The number of horizontal line segment in the n-th row equals A131576(n), the number of partitions of n into an even number of consecutive parts.
a(n) is the total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the n-th level of the diagram.
---------------------------------------------
   n    a(n)              Diagram
---------------------------------------------
   1      0
   2      0                               _
   3      2                              |2
   4      0                             _|1
   5      2                            |3
   6      0                           _|2
   7      2                          |4
   8      0                         _|3
   9      2                        |5     _
  10      4                       _|4    |4
  11      2                      |6      |3
  12      0                     _|5      |2
  13      2                    |7       _|1
  14      4                   _|6      |5
  15      2                  |8        |4
  16      0                 _|7        |3
  17      2                |9         _|2
  18      4               _|8        |6
  19      2              |10         |5
  20      0             _|9          |4   _
  21      8            |11          _|3  |6
  22      4           _|10         |7    |5
  23      2          |12           |6    |4
  24      0         _|11           |5    |3
  25      2        |13            _|4    |2
  26      4       _|12           |8     _|1
  27      8      |14             |7    |7
  28      0      |13             |6    |6
...
For n = 21 the number of horizontal line segment in the 21th row of the diagram equals A131576(21) = 2, the number of partitions of 21 into an even number of consecutive parts.
The total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the 21-th level of the diagram is equal to 2 + 6 = 8, so a(21) = 8.

Crossrefs

For more information about the diagram see A286000 and A237593.
Cf. A014105, A131576, A204217, A299765, A341309, A352505.

Formula

a(n) = A204217(n) - A341309(n), conjectured.
Conjecture: G.f.: Sum_{k>=1} 2*k*x^(k*(2*k+1))/(1-x^(2*k)). - Vaclav Kotesovec, Oct 23 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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