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

0,3

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

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

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)

Example

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

Crossrefs

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

Sequence in context: A117949 A228879 A010901 * A023624 A123194 A208668

Adjacent sequences: A187208 A187209 A187210 * A187212 A187213 A187214

Keyword

nonn,tabf

Author

Omar E. Pol, Mar 07 2011

Extensions

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

Status

approved