A233776 Number of grid points that are covered on the semi-infinite hexagonal grid after n-th stage in a cellular automaton in which the toothpicks are connected by their endpoints but the toothpicks placed in northwest or northeast direction are prohibited, starting with a(0) = 1.

1, 2, 4, 6, 9, 11, 15, 19, 24, 26, 30, 34, 40, 44, 52, 60, 69, 71, 75, 79, 85, 89, 97, 105, 115, 119, 127, 135, 147, 155, 171, 187, 204, 206, 210, 214, 220, 224, 232, 240, 250, 254, 262, 270, 282, 290, 306, 322, 340, 344, 352, 360, 372, 380, 396, 412, 432, 440, 456, 472, 496

Offset: 0

Author

Omar E. Pol, Dec 25 2013

Keywords

• nonn

Comments

a(n) is also the number of grid points that are covered on the semi-infinite square grid after n-th stage in a D-toothpick structure in which the toothpicks placed in horizontal direction are prohibited, starting with a(0) = 1.

Note that both the structure mentioned in the above comment and the structure mentioned in the definition are essentially equivalents. Both structures are related to the Sierpinski triangle (see A047999). The number of edges in the structure after n-th stage is A006046(n). The number of edges added at n-th stage is A001316(n-1), for n >= 1 (see example).

Examples

			Illustration of initial terms:
---------------------------------------------------------
. A001316 A006046         Diagram         A233777   a(n)
---------------------------------------------------------
.            0
.                            *               1       1
.    1       1               |
.                            *               1       2
.    2       3              / \
.                          *   *             2       4
.    2       5             |   |
.                          *   *             2       6
.    4       9            / \ / \
.                        *   *   *           3       9
.    2      11           |       |
.                        *       *           2      11
.    4      15          / \     / \
.                      *   *   *   *         4      15
.    4      19         |   |   |   |
.                      *   *   *   *         4      19
.    8      27        / \ / \ / \ / \
.                    *   *   *   *   *       5      24
.    2      29       |               |
.                    *               *       2      26
.    4      33      / \             / \
.                  *   *           *   *     4      30
.
		

Links

• Omar E. Pol, Illustration of the structure on the square grid after 32 stages
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Partial sums of A233777.

Cf. A001316, A006046, A047999, A139250, A141614, A160120, A160128, A160425, A194270, A194440, A233774.