A368516 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 and y != z.

2, 6, 4, 2, 10, 12, 8, 4, 2, 14, 20, 20, 12, 8, 4, 2, 18, 28, 32, 28, 18, 12, 8, 4, 2, 22, 36, 44, 44, 38, 24, 18, 12, 8, 4, 2, 26, 44, 56, 60, 58, 48, 32, 24, 18, 12, 8, 4, 2, 30, 52, 68, 76, 78, 72, 60, 40, 32, 24, 18, 12, 8, 4, 2, 34, 60, 80, 92, 98, 96

Offset

1,1

Comments

Row n consists of 2n-1 even positive integers.

Links

Table of n, a(n) for n=1..70.

Example

First six rows:
   2
   6    4    2
  10   12    8    4    2
  14   20   20   12    8    4    2
  18   28   32   28   18   12    8    4   2
  22   36   44   44   38   24   18   12   8   4   2
For n=3, there are 12 triples (x,y,z) having x != y and y != z:
  121:  |x-y| + |y-z| = 2
  123:  |x-y| + |y-z| = 2
  131:  |x-y| + |y-z| = 4
  132:  |x-y| + |y-z| = 3
  212:  |x-y| + |y-z| = 2
  213:  |x-y| + |y-z| = 3
  231:  |x-y| + |y-z| = 3
  232:  |x-y| + |y-z| = 2
  312:  |x-y| + |y-z| = 3
  313:  |x-y| + |y-z| = 4
  321:  |x-y| + |y-z| = 2
  323:  |x-y| + |y-z| = 2,
so that row 2 of the array is (6,4,2), representing six 2s, four 3s, and two 4s.

Mathematica

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

Crossrefs

Cf. A011379 (row sums), A007590 (limiting reverse row), A368434, A368437, A368515, A368517, A368518, A368519, A368520, A368521, A368522.

Sequence in context: A074382 A061350 A046276 * A283614 A333520 A003571

Adjacent sequences: A368513 A368514 A368515 * A368517 A368518 A368519

Keyword

nonn,tabf

Author

Clark Kimberling, Dec 31 2023

Status

approved