A187213 -id:A187213 - cp's OEIS frontend

A187212 Q-toothpick sequence in the first quadrant.

0, 1, 3, 5, 9, 13, 21, 31, 39, 43, 51, 63, 75, 91, 119, 149, 165, 169, 177, 189, 201, 217, 245, 277, 297, 313, 341, 377, 417, 477, 565, 643, 675, 679, 687, 699, 711, 727, 755, 787, 807, 823, 851, 887, 927, 987, 1075

Offset: 0

Omar E. Pol, Mar 22 2011, Mar 30 2011

nonn

At stage 0, we start with no Q-toothpicks.
At stage 1, we place a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1).
At stage 2, we place two Q-toothpicks.
The sequence gives the number of Q-toothpicks in the structure after n-th stage.
For more information see A187210.
A187213 gives the number of Q-toothpicks added at n-th stage.
Note that starting from (0,1), with the first Q-toothpick centered at (1,1), we have the toothpick sequence A139250.
Also, gullwing sequence on the semi-infinite square grid, since a "gull" is formed by two Q-toothpicks. The sequence gives the number of "gulls" (or G-toothpicks) in the structure after n-th stage. See A187220. - Omar E. Pol, Mar 30 2011

Nathaniel Johnston, Table of n, a(n) for n = 0..202
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

Cf. A139250, A160164, A187210, A187213, A187220.

It appears that a(n) = A139250(n) - 2*A059939(n), for n >= 1. - Omar E. Pol, Mar 29 2011

Terms after a(24) from Nathaniel Johnston, Mar 28 2011

A187217 Number of Q-toothpicks added at n-th stage to the structure of A187216.

Original entry on oeis.org

0, 2, 6, 8, 14, 22, 30, 22, 38, 54, 70, 22, 38, 54, 70, 54, 102, 150, 134, 22, 38, 54, 70, 54, 102, 150, 134, 54, 102, 150, 166, 182, 326, 406, 262, 22, 38, 54, 70, 54, 102, 150, 134, 54, 102, 150, 166, 182, 326, 406, 262, 54, 102, 150, 166, 182, 326, 406, 294, 182, 326

Offset: 0

Omar E. Pol, Mar 30 2011

nonn

Essentially the first differences of A187216.

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

Cf. A139250, A139251, A187216, A187211, A187213, A187221.

a(15) - a(60) from Nathaniel Johnston, Apr 15 2011

A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 4, 2, 4, 6, 6, 8, 14, 15, 8, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 39, 16, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 95

Offset: 1

Omar E. Pol, Mar 22 2011, Apr 06 2011

nonn

It appears that both a(2) and a(2^k - 1) are odd numbers, for k >= 2. Other terms are even numbers.

			At stage 1 we start in the first quadrant from a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1). There are no gulls in the structure, so a(1) = 0.
At stage 2 we place a gull (or G-toothpick) with its midpoint at (1,1) and its endpoints at (2,0) and (2,2), so a(2) = 1. There is only one exposed midpoint at (2,2).
At stage 3 we place a gull with its midpoint at (2,2), so a(3) = 1. There are two exposed endpoints.
At stage 4 we place two gulls, so a(4) = 2. There are two exposed endpoints.
At stage 5 we place two gulls, so a(5) = 2. There are four exposed endpoints.
And so on.
If written as a triangle begins:
0,
1,
1,2,
2,4,5,4,
2,4,6,6,8,14,15,8,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,39,16,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,42,28,...
It appears that rows converge to A151688.

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

Cf. A099035, A151550, A151688, A187210, A187211, A187212, A187213, A187220.

read("transforms3") ;
L := BFILETOLIST("b187212.txt") ;
A187213 := DIFF(L) ;
seq( op(n,A187213)/2,n=2..nops(A187213)) ; # R. J. Mathar, Mar 30 2011

a(1)=0. a(n) = A187213(n)/2, for n >= 2.

It appears that a(2^k - 1) = A099035(k-1), for k >= 2.

cp's OEIS Frontend

A187212 Q-toothpick sequence in the first quadrant.

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A187217 Number of Q-toothpicks added at n-th stage to the structure of A187216.

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A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

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