A183004 Toothpick sequence on square grid with toothpicks connected by their endpoints.

0, 1, 5, 11, 19, 27, 43, 65, 81, 89, 105, 129, 153, 185, 241, 303, 335, 343, 359, 383, 407, 439, 495, 559, 599, 631, 687, 759, 839, 959, 1135, 1293, 1357, 1365, 1381, 1405, 1429, 1461, 1517, 1581, 1621, 1653, 1709, 1781

Offset

0,3

Comments

Rules:

- If n is odd then each new toothpick must lie in vertical direction.

- If n is even then each new toothpick must lie in horizontal direction.

- Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.

The sequence gives the number of toothpicks after n stages. A183005 (the first differences) gives the number added at the n-th stage.

The structure is very similar to the structure of A139250 but the mechanism for the connection of toothpicks is different.

Links

Table of n, a(n) for n=0..43.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Formula

We start at stage 0 with no toothpicks.

At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis, so a(1)=1. There are two exposed endpoints.

At stage 2, place 4 toothpicks in horizontal position: two new toothpicks touching each exposed endpoint, so a(2)=1+4=5. There are 4 exposed endpoints.

At stage 3, place 6 toothpicks in vertical position, so a(3)=5+6=11.

After 3 stages the toothpick structure has 2 squares and 4 exposed endpoints.

Crossrefs

Cf. A139250, A160164, A182840, A183005.

Sequence in context: A190756 A022823 A094693 * A163120 A051350 A088638

Adjacent sequences: A183001 A183002 A183003 * A183005 A183006 A183007

Keyword

nonn

Author

Omar E. Pol, Mar 27 2011

Status

approved