A368517 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.

1, 1, 2, 4, 2, 1, 3, 7, 7, 4, 2, 1, 4, 10, 12, 11, 6, 4, 2, 1, 5, 13, 17, 18, 15, 9, 6, 4, 2, 1, 6, 16, 22, 25, 24, 20, 12, 9, 6, 4, 2, 1, 7, 19, 27, 32, 33, 31, 25, 16, 12, 9, 6, 4, 2, 1, 8, 22, 32, 39, 42, 42, 38, 31, 20, 16, 12, 9, 6, 4, 2, 1, 9, 25, 37

Offset

1,3

Comments

Row n consists of 2n positive integers.

Links

Table of n, a(n) for n=1..75.

Example

First eight rows:
  1   1
  2   4   2   1
  3   7   7   4   2   1
  4  10  12  11   6   4   2   1
  5  13  17  18  15   9   6   4   2   1
  6  16  22  25  24  20  12   9   6   4   2  1
  7  19  27  32  33  31  25  16  12   9   6  4  2  1
  8  22  32  39  42  42  38  31  20  16  12  9  6  4  2  1
For n=3, there are 9 triples (x,y,z) having x < y:
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  123:  |x-y| + |y-z| = 2
  131:  |x-y| + |y-z| = 4
  132:  |x-y| + |y-z| = 3
  133:  |x-y| + |y-z| = 2
  231:  |x-y| + |y-z| = 3
  232:  |x-y| + |y-z| = 2
  233:  |x-y| + |y-z| = 1,
so that row 2 of the array is (2,4,2,1), representing two 1s, four 2s, two 3s, and one 4.

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. A006002 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368518, A368519, A368520, A368521, A368522.

Sequence in context: A356474 A079045 A021417 * A355346 A364608 A105791

Adjacent sequences: A368514 A368515 A368516 * A368518 A368519 A368520

Keyword

nonn,tabf

Author

Clark Kimberling, Dec 31 2023

Status

approved