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

1,1

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).

Links

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

Example

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:
  + - - +     + * * +  + * - +  + * - +  + * - +  + * - +  + * - +  + * - +
  |     | --\ |     |  |     *  |     |  |     |  |     |  |     |  *     |
  |     | --/ |     |  |     |  |     *  |     |  |     |  *     |  |     |
  + - - +     + - - +, + - - +, + - - +, + - * +, + * - +, + - - +, + - - +.

Crossrefs

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

Sequence in context: A047552 A287453 A287449 * A369648 A221847 A283766

Adjacent sequences: A287685 A287686 A287687 * A287689 A287690 A287691

Keyword

nonn,tabl

Author

Doug Bell, May 29 2017

Status

approved