A368608 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, 1, 4, 5, 2, 1, 6, 9, 8, 4, 2, 1, 8, 13, 14, 12, 6, 4, 2, 1, 10, 17, 20, 20, 16, 9, 6, 4, 2, 1, 12, 21, 26, 28, 26, 21, 12, 9, 6, 4, 2, 1, 14, 25, 32, 36, 36, 33, 26, 16, 12, 9, 6, 4, 2, 1, 16, 29, 38, 44, 46, 45, 40, 32, 20, 16, 12, 9, 6, 4, 2, 1, 18, 33

Offset

1,1

Comments

Row n consists of 2n positive integers.

Links

Table of n, a(n) for n=1..74.

Example

First six rows:
   2   1
   4   5   2   1
   6   9   8   4   2   1
   8  13  14  12   6   4   2  1
  10  17  20  20  16   9   6  4  2  1
  12  21  26  28  26  21  12  9  6  4  2  1
For n=2, there are 3 triples (x,y,z) having x != y and y <= z:
  122:  |x-y| + |y-z| = 1
  211:  |x-y| + |y-z| = 1
  212:  |x-y| + |y-z| = 2
so row 2 of the array is (2,1), representing two 1s and one 2.

Mathematica

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[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, 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. A005443 (column 1), A027480 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368607, A368609.

Sequence in context: A370181 A248666 A162407 * A352458 A132741 A072436

Adjacent sequences: A368605 A368606 A368607 * A368609 A368610 A368611

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 25 2024

Status

approved