A368952 Irregular triangle T(n,k) read by rows: row n lists the larger number in each pair of triangular numbers (a, b) satisfying a - b = n.

1, 3, 6, 3, 10, 15, 6, 21, 6, 28, 10, 36, 45, 15, 10, 55, 10, 66, 21, 78, 15, 91, 28, 105, 15, 120, 36, 21, 15, 136, 153, 45, 171, 28, 21, 190, 55, 210, 21, 231, 66, 36, 21, 253, 28, 276, 78, 300, 45, 325, 91, 28, 351, 36, 378, 105, 55, 28, 406, 28, 435, 120, 465, 66, 45, 36

Offset: 1

Author

Hartmut F. W. Hoft, Jan 10 2024

Keywords

• nonn
• tabf

Comments

The length of row n in the triangle is A001227(n) and its first column T(n, 1) is ordered. Also, A001227(n) = number of 1s in row n of the triangle of A237048 = length of row n in the triangle of A280851. The records of row lengths in the triangle form sequence A038547.

Examples

			For n=3 with 0 <= k <= 6, sqrt((2*k + 1)^2 - 8*3) has integer values for k=2, 3, so that the pairs of triangular numbers are (3, 0) and (6, 3), and row 3 of the triangle consists of 6 and 3.
The first 20 rows of the irregular triangle:
   n| k:   1     2     3     4
  -----------------------------
   1|      1
   2|      3
   3|      6     3
   4|     10
   5|     15     6
   6|     21     6
   7|     28    10
   8|     36
   9|     45    15    10
  10|     55    10
  11|     66    21
  12|     78    15
  13|     91    28
  14|    105    15
  15|    120    36    21    15
  16|    136
  17|    153    45
  18|    171    28    21
  19|    190    55
  20|    210    21
  ...
		

Crossrefs

Cf. A000217, A001227, A038547, A136107, A237048, A280851.

Programs

Mathematica

a000217[k_] := k (k+1)/2
triangle[n_] := Map[a000217, Select[Range[a000217[n], 0, -1], IntegerQ[Sqrt[(2#+1)^2 -8n]]&]]
a368952[n_] := Flatten[Map[triangle, Range[n]]]
a368952[30]

Formula

n = A000217(x) - A000217(y), x > y >= 0, precisely when sqrt( (2*x + 1)^2 - 8*n ) is an integer.