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A368515 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.

%I A368515 #7 Jan 20 2024 09:46:05
%S A368515 2,2,4,8,4,2,6,14,14,8,4,2,8,20,24,22,12,8,4,2,10,26,34,36,30,18,12,8,
%T A368515 4,2,12,32,44,50,48,40,24,18,12,8,4,2,14,38,54,64,66,62,50,32,24,18,
%U A368515 12,8,4,2,16,44,64,78,84,84,76,62,40,32,24,18,12,8,4
%N A368515 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.
%C A368515 Row n consists of 2n even positive integers.
%e A368515 First six rows:
%e A368515   2    2
%e A368515   4    8   4     2
%e A368515   6   14   14    8    4   2
%e A368515   8   20   24   22   12   8     4    2
%e A368515  10   26   34   36   30   18   12    8    4   2
%e A368515  12   32   44   50   48   40   24   18   12   8   4   2
%e A368515 For n=2, there are 4 triples (x,y,z) having x != y:
%e A368515   121:  |x-y| + |y-z| = 2
%e A368515   122:  |x-y| + |y-z| = 1
%e A368515   211:  |x-y| + |y-z| = 1
%e A368515   212:  |x-y| + |y-z| = 2,
%e A368515 so that row 2 of the array is (2,2), representing two 1s and two 2s.
%t A368515 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368515 t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] &];
%t A368515 a[n_, k_] :=   Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368515 u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
%t A368515 v = Flatten[u]; (* sequence *)
%t A368515 Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]]  (* array *)
%Y A368515 Cf. A045991 (row sums), A007590 (limiting reverse row), A368434, A368437, A368516, A368517, A368518, A368519, A368520, A368521, A368522.
%K A368515 nonn,tabf
%O A368515 1,1
%A A368515 _Clark Kimberling_, Dec 31 2023