A183005 -id:A183005 - cp's OEIS frontend

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

Omar E. Pol, Mar 27 2011

nonn

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.

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

Cf. A139250, A160164, A182840, A183005.

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula