A368437 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = 2n+1-k, where (x,y,z) is a permutation of three distinct numbers taken from {0,1,...,n}, for n >= 2, k >= 2.

4, 2, 4, 4, 12, 4, 4, 4, 12, 14, 20, 6, 4, 4, 12, 12, 28, 24, 28, 8, 4, 4, 12, 12, 24, 30, 44, 34, 36, 10, 4, 4, 12, 12, 24, 24, 48, 48, 60, 44, 44, 12, 4, 4, 12, 12, 24, 24, 40, 50, 72, 66, 76, 54, 52, 14, 4, 4, 12, 12, 24, 24, 40, 40, 72, 76, 96, 84, 92

Offset: 1

Author

Clark Kimberling, Dec 25 2023

Keywords

• nonn
• tabf

Comments

Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!. The limiting row, (4, 4, 12, 12, 24, 24, 40, 40, ...) consists of repeated terms of (A046092(n+1)) = (4, 12, 24, 40, ...).

Examples

			Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = 2n+1-k, for k = 2,3, are as follows:
012: |0-1| + |1-2| = 2
021: |0-2| + |2-1| = 3
102: |1-0| + |0-2| = 3
120: |1-2| + |2-0| = 3
201: |2-0| + |0-1| = 3
210: |2-1| + |1-0| = 2
so that row 1 of the array is (4,2), representing four 2s and two 3s.
First eight rows:
  4  2
  4  4  12   4
  4  4  12  14  20   6
  4  4  12  12  28  24  28   8
  4  4  12  12  24  30  44  34  36  10
  4  4  12  12  24  24  48  48  60  44  44  12
  4  4  12  12  24  24  40  50  72  66  76  54  52  14
  4  4  12  12  24  24  40  40  72  76  96  84  92  64  60  16
		

Crossrefs

Cf. A007531, A046092, A368435, A368436.

Programs

Mathematica

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