A368952 - OEIS

%I #11 Jun 03 2024 18:40:36

%S 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 15,120,36,21,15,136,153,45,171,28,21,190,55,210,21,231,66,36,21,253,

%U 28,276,78,300,45,325,91,28,351,36,378,105,55,28,406,28,435,120,465,66,45,36

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

%e 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 The first 20 rows of the irregular triangle:

%e    n| k:   1     2     3     4

%e   -----------------------------

%e    1|      1

%e    2|      3

%e    3|      6     3

%e    4|     10

%e    5|     15     6

%e    6|     21     6

%e    7|     28    10

%e    8|     36

%e    9|     45    15    10

%e   10|     55    10

%e   11|     66    21

%e   12|     78    15

%e   13|     91    28

%e   14|    105    15

%e   15|    120    36    21    15

%e   16|    136

%e   17|    153    45

%e   18|    171    28    21

%e   19|    190    55

%e   20|    210    21

%e   ...

%t a000217[k_] := k (k+1)/2

%t triangle[n_] := Map[a000217, Select[Range[a000217[n], 0, -1], IntegerQ[Sqrt[(2#+1)^2 -8n]]&]]

%t a368952[n_] := Flatten[Map[triangle, Range[n]]]

%t a368952[30]

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

%K nonn,tabf

%O 1,2

%A _Hartmut F. W. Hoft_, Jan 10 2024