A187211 First differences of A187210.

0, 1, 4, 7, 12, 22, 20, 22, 40, 54, 40, 22, 40, 54, 56, 70, 120, 134, 72, 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136, 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 152, 70, 120, 150, 168, 246, 360, 342, 232, 246, 376, 454, 568, 838, 1032

Offset: 0

Author

Omar E. Pol, Mar 07 2011

Keywords

• nonn
• tabf

Comments

Number of Q-toothpicks added at n-th stage to the Q-toothpick structure of A187210.

For the connection with A139251, the first differences of the toothpick sequence A139250, see the Formula section. - Omar E. Pol, Apr 02 2016

Examples

			Written as an irregular triangle the sequence begins:
0;
1;
4;
7;
12;
22, 20;
22, 40, 54, 40;
22, 40, 54, 56, 70, 120, 134, 72;
22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136;
...
The rows of this triangle tend to A188156.
From _Omar E. Pol_, Apr 02 2016: (Start)
For n = 5 we have that A139251(5-2) = 4, A267699(5-2) = 7 and A267695(5-1) = 7, so a(5) = 2*4 + 7 + 7 = 22.
For n = 10 we have that A139251(10-2) = 8, A267699(10-2) = 20 and A267695(10-1) = 4, so a(10) = 2*8 + 20 + 4 = 40.
(End)
Starting from a(3) = 7 the row lengths of triangle are the terms of A011782. - _Omar E. Pol_, Apr 04 2016
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..177
• David Applegate, The movie version
• 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, C script
• Nathaniel Johnston, The Q-Toothpick Cellular Automaton
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Crossrefs

Cf. A011782, A139250, A139251, A172472, A182841, A187210, A187221, A267695, A267699.

Formula

a(2^n + 2) = 16 + 8(2^(n-1) - 1), n >= 3. [Nathaniel Johnston, Mar 26 2011]

From Omar E. Pol, Apr 02 2016: (Start)

a(n) = floor(sqrt(2*n^3)), if 0<=n<=2 or n=6.

a(n) = 2*A139251(n-2) + A267699(n-2) + A267695(n-1), if 3<=n<=5 or n>=7.

(End)

Extensions

Terms after a(7) from Nathaniel Johnston, Mar 26 2011