cp's OEIS Frontend

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.

%I A368952 #11 Jun 03 2024 18:40:36
%S A368952 1,3,6,3,10,15,6,21,6,28,10,36,45,15,10,55,10,66,21,78,15,91,28,105,
%T A368952 15,120,36,21,15,136,153,45,171,28,21,190,55,210,21,231,66,36,21,253,
%U A368952 28,276,78,300,45,325,91,28,351,36,378,105,55,28,406,28,435,120,465,66,45,36
%N 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.
%C A368952 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.
%F A368952 n = A000217(x) - A000217(y), x > y >= 0, precisely when sqrt( (2*x + 1)^2 - 8*n ) is an integer.
%e A368952 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.
%e A368952 The first 20 rows of the irregular triangle:
%e A368952    n| k:   1     2     3     4
%e A368952   -----------------------------
%e A368952    1|      1
%e A368952    2|      3
%e A368952    3|      6     3
%e A368952    4|     10
%e A368952    5|     15     6
%e A368952    6|     21     6
%e A368952    7|     28    10
%e A368952    8|     36
%e A368952    9|     45    15    10
%e A368952   10|     55    10
%e A368952   11|     66    21
%e A368952   12|     78    15
%e A368952   13|     91    28
%e A368952   14|    105    15
%e A368952   15|    120    36    21    15
%e A368952   16|    136
%e A368952   17|    153    45
%e A368952   18|    171    28    21
%e A368952   19|    190    55
%e A368952   20|    210    21
%e A368952   ...
%t A368952 a000217[k_] := k (k+1)/2
%t A368952 triangle[n_] := Map[a000217, Select[Range[a000217[n], 0, -1], IntegerQ[Sqrt[(2#+1)^2 -8n]]&]]
%t A368952 a368952[n_] := Flatten[Map[triangle, Range[n]]]
%t A368952 a368952[30]
%Y A368952 Cf. A000217, A001227, A038547, A136107, A237048, A280851.
%K A368952 nonn,tabf
%O A368952 1,2
%A A368952 _Hartmut F. W. Hoft_, Jan 10 2024