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A183004
Toothpick sequence on square grid with toothpicks connected by their endpoints.
3
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
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.]
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Mar 27 2011
STATUS
approved