A287618 Triangle read by rows: T(j,k) is the number of distinct edge segments in a j X k rectangular grid.

1, 2, 1, 3, 3, 2, 3, 3, 4, 2, 4, 4, 5, 5, 3, 4, 4, 5, 5, 6, 3, 5, 5, 6, 6, 7, 7, 4, 5, 5, 6, 6, 7, 7, 8, 4, 6, 6, 7, 7, 8, 8, 9, 9, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 5, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 6, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 7

Offset: 1

Author

Doug Bell, May 28 2017

Keywords

• nonn
• tabl
• hear

Comments

This gives the number of edge segments that are distinct with respect to rotation and mirror images. Sequence is arranged so that j <= k (since 2 X 3 and 3 X 2 are equivalent grids), first by increasing j, then by increasing k: a(1) = 1 X 1 = 1, a(2) = 1 X 2 = 2, a(3) = 2 X 2 = 1, a(4) = 1 X 3 = 3.

Here j = A002260(n), k = A002024(n), and n = A000217(k-1) + j, then a(n) = if j = k, ceiling(j/2), else ceiling(j/2) + ceiling(k/2).

Examples

			Triangle begins:
  1;
  2, 1;
  3, 3, 2;
  3, 3, 4, 2;
  4, 4, 5, 5, 3;
  4, 4, 5, 5, 6, 3;
  5, 5, 6, 6, 7, 7, 4;
...
For n = 9, the a(9) = 4 distinct edge segments [A,B,C,D] for a 3 X 4 rectangular grid are:
  + - - - - +       + A B B A +
  |         |       C         C
  |         |  -->  D         D
  |         |       C         C
  + - - - - +       + A B B A +.
		

Links

• Doug Bell, Table of n, a(n) for n = 1..11325, Rows n = 1..150, flattened.

Crossrefs

Cf. A002260, A002024, A000217.

Cf. A287688 (number of distinct edge segment pairs).

Programs

Mathematica

Table[Ceiling[j/2] + Boole[j != k] Ceiling[k/2], {j, 14}, {k, j}] // Flatten (* Michael De Vlieger, Jun 09 2017 *)