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.

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, 4, 2, 12, 32, 44, 50, 48, 40, 24, 18, 12, 8, 4, 2, 14, 38, 54, 64, 66, 62, 50, 32, 24, 18, 12, 8, 4, 2, 16, 44, 64, 78, 84, 84, 76, 62, 40, 32, 24, 18, 12, 8, 4

Offset

1,1

Comments

Row n consists of 2n even positive integers.

Links

Table of n, a(n) for n=1..71.

Example

First six rows:
  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    4   2
 12   32   44   50   48   40   24   18   12   8   4   2
For n=2, there are 4 triples (x,y,z) having x != y:
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  211:  |x-y| + |y-z| = 1
  212:  |x-y| + |y-z| = 2,
so that row 2 of the array is (2,2), representing two 1s and two 2s.

Mathematica

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] &];
a[n_, k_] :=   Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
v = Flatten[u]; (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]]  (* array *)

Crossrefs

Cf. A045991 (row sums), A007590 (limiting reverse row), A368434, A368437, A368516, A368517, A368518, A368519, A368520, A368521, A368522.

Sequence in context: A045777 A136534 A121175 * A366714 A359488 A183397

Adjacent sequences: A368512 A368513 A368514 * A368516 A368517 A368518

Keyword

nonn,tabf

Author

Clark Kimberling, Dec 31 2023

Status

approved