A368434 - OEIS

%I #5 Dec 30 2023 23:41:21

%S 1,2,4,2,3,8,10,4,2,4,12,18,16,8,4,2,5,16,26,28,24,12,8,4,2,6,20,34,

%T 40,40,32,18,12,8,4,2,7,24,42,52,56,52,42,24,18,12,8,4,2,8,28,50,64,

%U 72,72,66,52,32,24,18,12,8,4,2,9,32,58,76,88,92,90,80

%N Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {0,1,...,n}.

%C Row n consists of 2n-1 positive integers having sum A000575(n) = n^3.

%e First eight rows:

%e 1

%e 2   4   2

%e 3   8  10   4   2

%e 4  12  18  16   8   4   2

%e 5  16  26  28  24  12   8   4   2

%e 6  20  34  40  40  32  18  12   8   4   2

%e 7  24  42  52  56  52  42  24  18  12   8   4  2

%e 8  28  50  64  72  72  66  52  32  24  18  12  8  4  2

%t t[n_] := t[n] = Tuples[Range[n], 3];

%t a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];

%t u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];

%t Flatten[u] (* sequence *)

%t Column[u]  (* array *)

%Y Cf. A000575, A007590 (limiting reversed row), A368435 (reversed rows), A368435, A368346.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Dec 25 2023