A267699 -id:A267699 - cp's OEIS frontend

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

Omar E. Pol, Mar 07 2011

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

			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

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

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

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)

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

A267698 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,3) and (3,1) respectively. The endpoints of the left hand Q-toothpick are at (0,3) and (1,4). The endpoints of the right hand Q-toothpick are at (3,0) and (4,1). With a(0) = 0.

Original entry on oeis.org

0, 2, 6, 13, 25, 32, 44, 59, 79, 86, 98, 113, 133, 148, 176, 215, 251, 258, 270, 285, 305, 320, 348, 387, 423, 438

Offset: 0

Omar E. Pol, Jan 23 2016

a(n) is the total number of Q-toothpicks in the structure after n-th stage.
A267699 (the first differences) gives the number of Q-toothpicks added at n-th stage.
Note that this sequence is also related with the structure and the formula of A187210. For more information see A187210, A267694 and A139250.

Cf. A139250, A187210, A187220, A267694, A267699.

A267695 First differences of A267694.

Original entry on oeis.org

0, 2, 3, 4, 7, 4, 7, 12, 15, 4, 7, 12, 15, 12, 23, 36, 31, 4, 7, 12, 15, 12, 23, 36, 31, 12

Offset: 0

Omar E. Pol, Apr 02 2016

Number of Q-toothpicks added at n-th stage to the Q-toothpick structure of A267694.

			When the positive terms are written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
2;
3;
4, 7;
4, 7, 12, 15;
4, 7, 12, 15, 12, 23, 36, 31;
4, 7, 12, 15, 12, 23, 36, 31, 12,...

Cf. A011782, A187210, A187211, A267694, A267699.

cp's OEIS Frontend

A187211 First differences of A187210.

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A267698 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,3) and (3,1) respectively. The endpoints of the left hand Q-toothpick are at (0,3) and (1,4). The endpoints of the right hand Q-toothpick are at (3,0) and (4,1). With a(0) = 0.

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A267695 First differences of A267694.

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