A287688 Triangle read by rows: T(j,k) is the number of distinct edge segment pairs in a j X k rectangular grid.

2, 6, 7, 10, 15, 13, 15, 21, 28, 22, 21, 28, 36, 45, 32, 28, 36, 45, 55, 66, 45, 36, 45, 55, 66, 78, 91, 59, 45, 55, 66, 78, 91, 105, 120, 76, 55, 66, 78, 91, 105, 120, 136, 153, 94, 66, 78, 91, 105, 120, 136, 153, 171, 190, 115, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 137

Offset: 1

Author

Doug Bell, May 29 2017

Keywords

• nonn
• tabl

Comments

This gives the number of pairs 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 = 2, a(2) = 1 X 2 = 6, a(3) = 2 X 2 = 7, a(4) = 1 X 3 = 10.

Here j = A002260(n), k = A002024(n), and n = A000217(k-1) + j.

Where j != k, a(n) = A000217(j + k).

Where j = k, a(n) is approximately A236312(j-2); a(n) >= A236312(j-2).

Examples

			Triangle begins:
   2;
   6,  7;
  10, 15, 13;
  15, 21, 28, 22;
  21, 28, 36, 45, 32;
  28, 36, 45, 55, 66, 45;
  36, 45, 55, 66, 78, 91, 59;
...
For n = 3, the a(3) = 7 pairs of edge segments for a 2 X 2 rectangular grid are:
  + - - +     + * * +  + * - +  + * - +  + * - +  + * - +  + * - +  + * - +
  |     | --\ |     |  |     *  |     |  |     |  |     |  |     |  *     |
  |     | --/ |     |  |     |  |     *  |     |  |     |  *     |  |     |
  + - - +     + - - +, + - - +, + - - +, + - * +, + * - +, + - - +, + - - +.
		

Links

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

Crossrefs

Cf. A002260, A002024, A000217, A236312. Distinct edge segments A287618.