A187216 Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.

0, 2, 8, 16, 30, 52, 82, 104, 142, 196, 266, 288, 326, 380, 450, 504, 606, 756, 890, 912, 950, 1004, 1074, 1128, 1230, 1380, 1514, 1568, 1670, 1820, 1986, 2168, 2494, 2900, 3162, 3184, 3222, 3276, 3346, 3400, 3502, 3652, 3786, 3840, 3942, 4092, 4258, 4440

Offset

0,2

Comments

The sequence gives the number of Q-toothpicks in the structure after n-th stage.

A187217 (the first differences) gives the number of Q-toothpicks added at n-th stage.

Note that in the Q-toothpick structure sometimes there is also an internal growth of Q-toothpicks.

For more information see A187210.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..200

Nathaniel Johnston, C program for computing terms

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.]

Nathaniel Johnston, The Q-Toothpick Cellular Automaton

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

Example

On the infinite square grid at stage 0 we start with no Q-toothpicks.
At stage 1 we place two opposite Q-toothpicks centered at (0,0). One of the Q-toothpicks lies on the first quadrant with its endpoints at (0,1) and (1,0). The other Q-toothpick lies on the third quadrant with its endpoints at (0,-1) and (-1,0). So a(1) = 2. There are 4 exposed endpoints.
At stage 2 we place 6 Q-toothpicks, so a(2) = 2+6 = 8.
At stage 3 we place 8 Q-toothpicks, so a(3) = 8+8 = 16.
At stage 4 we place 14 Q-toothpicks, so a(4) = 16+14 = 30.
After 4 stages in the Q-toothpick structure there are 1 circle, 2 "heads" and 12 exposed endpoints.

Crossrefs

Cf. A139250, A160120, A160164, A187210, A187212, A187217, A187220.

Sequence in context: A327329 A136514 A077071 * A210729 A294534 A294542

Adjacent sequences: A187213 A187214 A187215 * A187217 A187218 A187219

Keyword

nonn

Author

Omar E. Pol, Mar 30 2011

Extensions

a(15) - a(47) from Nathaniel Johnston, Apr 15 2011

Status

approved