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A187216
Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.
2
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, 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.]
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.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Mar 30 2011
EXTENSIONS
a(15) - a(47) from Nathaniel Johnston, Apr 15 2011
STATUS
approved