A338721 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the odd numbers k times, and the first element of column k is in row k(k+1)/2.

1, 3, 5, 1, 7, 1, 9, 3, 11, 3, 1, 13, 5, 1, 15, 5, 1, 17, 7, 3, 19, 7, 3, 1, 21, 9, 3, 1, 23, 9, 5, 1, 25, 11, 5, 1, 27, 11, 5, 3, 29, 13, 7, 3, 1, 31, 13, 7, 3, 1, 33, 15, 7, 3, 1, 35, 15, 9, 5, 1, 37, 17, 9, 5, 1, 39, 17, 9, 5, 3, 41, 19, 11, 5, 3, 1, 43, 19, 11, 7, 3, 1

Offset

1,2

Comments

A missing companion to A196020 and A235791.

T(n,k) is the total number of horizontal steps in the first n levels of the k-th largest double-staircase of the diagram defined in A335616 (see example). - Omar E. Pol, Nov 30 2020

Column k is the partial sums of the k-th column of A339275. - Omar E. Pol, Dec 01 2020

Links

Alois P. Heinz, Rows n = 1..500, flattened

Formula

T(n,k) = 2 * floor((n-k*(k-1)/2)/k) - 1. - Alois P. Heinz, Nov 30 2020

Example

Triangle begins:
   1;
   3;
   5,  1;
   7,  1;
   9,  3;
  11,  3,  1;
  13,  5,  1;
  15,  5,  1;
  17,  7,  3;
  19,  7,  3,  1;
  21,  9,  3,  1;
  23,  9,  5,  1;
  25, 11,  5,  1;
  27, 11,  5,  3;
  29, 13,  7,  3,  1;
  31, 13,  7,  3,  1;
  33, 15,  7,  3,  1;
  35, 15,  9,  5,  1;
  37, 17,  9,  5,  1;
  39, 17,  9,  5,  3;
  41, 19, 11,  5,  3,  1;
  43, 19, 11,  7,  3,  1;
  45, 21, 11,  7,  3,  1;
  47, 21, 13,  7,  3,  1;
  49, 23, 13,  7,  5,  1;
  51, 23, 13,  9,  5,  1;
  53, 25, 15,  9,  5,  3;
  55, 25, 15,  9,  5,  3,  1;
...
From Omar E. Pol, Nov 30 2020: (Start)
For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
For n = 15 the diagram with first 15 levels looks like this:
.
Level                         "Double-staircases" diagram
.                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
The first largest double-staircase has 29 horizontal steps, the second double-staircase has 13 steps, the third double-staircase has 7 steps, the fourth double-staircase has 3 steps and the fifth double-staircase has only one step, so the 15th row of triangle is [29, 13, 7, 3, 1]. (End)

Maple

T:= (n, k)-> 2*iquo(n-k*(k-1)/2, k)-1:
seq(seq(T(n, k), k=1..floor((sqrt(1+8*n)-1)/2)), n=1..30);  # Alois P. Heinz, Nov 30 2020

Crossrefs

Cf. A196020, A235791, A237593, A335616, A338722, A338723, A339275.

Sequence in context: A196020 A346775 A028510 * A122053 A124084 A133045

Adjacent sequences: A338718 A338719 A338720 * A338722 A338723 A338724

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 30 2020.

Status

approved