A368434 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,...,n}.

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, 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, 72, 72, 66, 52, 32, 24, 18, 12, 8, 4, 2, 9, 32, 58, 76, 88, 92, 90, 80, 64, 40, 32, 24, 18, 12, 8, 4, 2

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

From Alois P. Heinz, Nov 25 2025: (Start)

Sum_{k>0} k * T(n,k) = A072819(n).

Sum_{k>=0} (-1)^k * T(n,k) = A193356(n). (End)

Example

First eight rows:
  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  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  72  72  66  52  32  24  18  12  8  4  2
  ...

Mathematica

t[n_] := t[n] = Tuples[Range[n], 3];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == 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 reversed row), A368435 (reversed rows), A368346.

Cf. A072819, A193356.

Sequence in context: A141387 A349400 A335678 * A134400 A016095 A298309

Adjacent sequences: A368431 A368432 A368433 * A368435 A368436 A368437

Keyword

nonn,tabf

Author

Clark Kimberling, Dec 25 2023

Status

approved