A368607 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.

1, 3, 2, 1, 5, 6, 4, 2, 1, 7, 10, 10, 6, 4, 2, 1, 9, 14, 16, 14, 9, 6, 4, 2, 1, 11, 18, 22, 22, 19, 12, 9, 6, 4, 2, 1, 13, 22, 28, 30, 29, 24, 16, 12, 9, 6, 4, 2, 1, 15, 26, 34, 38, 39, 36, 30, 20, 16, 12, 9, 6, 4, 2, 1, 17, 30, 40, 46, 49, 48, 44, 36, 25

Offset

1,2

Comments

Row n consists of 2n-1 positive integers.

Links

Table of n, a(n) for n=1..73.

Example

First six rows:
 1
 3    2   1
 5    6   4   2  1
 7   10  10   6  4   2   1
 9   14  16  14  9   6   4  2  1
 11  18  22  22  19  12  9  6  4  2  1
For n=3, there are 6 triples (x,y,z) having x != y and y < z:
  123:  |x-y| + |y-z| = 2
  212:  |x-y| + |y-z| = 2
  213:  |x-y| + |y-z| = 3
  312:  |x-y| + |y-z| = 3
  313:  |x-y| + |y-z| = 4
  323:  |x-y| + |y-z| = 2
so row 2 of the array is (3,2,1), representing three 2s, two 3s, and one 4.

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, 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. A005408 (column 1), A002411 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368609.

Sequence in context: A208608 A209577 A139377 * A138483 A110712 A065366

Adjacent sequences: A368604 A368605 A368606 * A368608 A368609 A368610

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 25 2024

Status

approved