A368605 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, 1, 2, 3, 2, 1, 3, 5, 5, 4, 2, 1, 4, 7, 8, 8, 6, 4, 2, 1, 5, 9, 11, 12, 11, 9, 6, 4, 2, 1, 6, 11, 14, 16, 16, 15, 12, 9, 6, 4, 2, 1, 7, 13, 17, 20, 21, 21, 19, 16, 12, 9, 6, 4, 2, 1, 8, 15, 20, 24, 26, 27, 26, 24, 20, 16, 12, 9, 6, 4, 2, 1, 9, 17, 23, 28

Offset

1,3

Comments

Row n consists of 2n positive integers.

Links

Table of n, a(n) for n=1..76.

Example

First six rows:
  1    1
  2    3    2    1
  3    5    5    4    2    1
  4    7    8    8    6    4    2   1
  5    9   11   12   11    9    6   4   2   1
  6   11   14   16   16   15   12   9   6   4   2   1
For n=3, there are 8 triples (x,y,z) having x < y and y >= z:
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  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 row 1 of the array is (2,3,2,1), representing two 1s, 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, 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. A000027 (column 1), A007290 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368606, A368607, A368609.

Sequence in context: A107357 A251718 A182457 * A334715 A026105 A303868

Adjacent sequences: A368602 A368603 A368604 * A368606 A368607 A368608

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 22 2024

Status

approved