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.
%I A368605 #15 Mar 29 2025 00:23:18
%S A368605 1,1,2,3,2,1,3,5,5,4,2,1,4,7,8,8,6,4,2,1,5,9,11,12,11,9,6,4,2,1,6,11,
%T A368605 14,16,16,15,12,9,6,4,2,1,7,13,17,20,21,21,19,16,12,9,6,4,2,1,8,15,20,
%U A368605 24,26,27,26,24,20,16,12,9,6,4,2,1,9,17,23,28
%N A368605 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x < y and y >= z.
%C A368605 Row n consists of 2n positive integers.
%e A368605 First six rows:
%e A368605 1 1
%e A368605 2 3 2 1
%e A368605 3 5 5 4 2 1
%e A368605 4 7 8 8 6 4 2 1
%e A368605 5 9 11 12 11 9 6 4 2 1
%e A368605 6 11 14 16 16 15 12 9 6 4 2 1
%e A368605 For n=3, there are 8 triples (x,y,z) having x < y and y >= z:
%e A368605 121: |x-y| + |y-z| = 2
%e A368605 122: |x-y| + |y-z| = 1
%e A368605 131: |x-y| + |y-z| = 4
%e A368605 132: |x-y| + |y-z| = 3
%e A368605 133: |x-y| + |y-z| = 2
%e A368605 231: |x-y| + |y-z| = 3
%e A368605 232: |x-y| + |y-z| = 2
%e A368605 233: |x-y| + |y-z| = 1
%e A368605 so row 1 of the array is (2,3,2,1), representing two 1s, three 2s, two 3s, and one 4.
%t A368605 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368605 t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] >= #[[3]] &];
%t A368605 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368605 u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
%t A368605 v = Flatten[u] (* sequence *)
%t A368605 Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]] (* array *)
%Y A368605 Cf. A000027 (column 1), A007290 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368606, A368607, A368609.
%K A368605 nonn,tabf
%O A368605 1,3
%A A368605 _Clark Kimberling_, Jan 22 2024