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

1,2

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).

Links

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

Example

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 +.

Mathematica

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

Crossrefs

Cf. A002260, A002024, A000217.

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

Sequence in context: A350743 A234200 A102746 * A123143 A128133 A032434

Adjacent sequences: A287615 A287616 A287617 * A287619 A287620 A287621

Keyword

nonn,tabl,hear

Author

Doug Bell, May 28 2017

Status

approved