A368435 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = 2n-2-k, where x,y,z are in {1,...,n}.

1, 2, 4, 2, 2, 4, 10, 8, 3, 2, 4, 8, 16, 18, 12, 4, 2, 4, 8, 12, 24, 28, 26, 16, 5, 2, 4, 8, 12, 18, 32, 40, 40, 34, 20, 6, 2, 4, 8, 12, 18, 24, 42, 52, 56, 52, 42, 24, 7, 2, 4, 8, 12, 18, 24, 32, 52, 66, 72, 72, 64, 50, 28, 8, 2, 4, 8, 12, 18, 24, 32, 40, 64, 80, 90, 92, 88, 76, 58, 32, 9

Offset

1,2

Comments

Row n consists of 2n-1 positive integers having sum A000578(n) = n^3.

Links

Alois P. Heinz, Rows n = 1..100, flattened

Formula

Sum_{k>=0} (-1)^k * T(n,k) = A193356(n). - Alois P. Heinz, Nov 25 2025

Example

First eight rows:
  1
  2  4   2
  2  4  10   8   3
  2  4   8  16  18  12   4
  2  4   8  12  24  28  26  16   5
  2  4   8  12  18  32  40  40  34  20   6
  2  4   8  12  18  24  42  52  56  52  42  24   7
  2  4   8  12  18  24  32  52  66  72  72  64  50  28  8
  ...

Mathematica

t[n_] := t[n] = Tuples[Range[n], 3];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == 2n-2-k &];
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];
Flatten[u]  (* sequence *)
Column[u]   (* array *)

Crossrefs

Cf. A000578, A007590 (limiting row), A368434, A368437.

Cf. A193356.

Sequence in context: A213433 A294354 A144049 * A058384 A255671 A330590

Adjacent sequences: A368432 A368433 A368434 * A368436 A368437 A368438

Keyword

nonn,tabf

Author

Clark Kimberling, Dec 25 2023

Status

approved