A187220 -id:A187220 - cp's OEIS frontend

A160164 Number of toothpicks after n-th stage in the I-toothpick structure of A139250.

0, 2, 6, 14, 22, 30, 46, 70, 86, 94, 110, 134, 158, 190, 246, 310, 342, 350, 366, 390, 414, 446, 502, 566, 606, 638, 694, 766, 846, 966, 1142, 1302, 1366, 1374, 1390, 1414, 1438, 1470, 1526, 1590, 1630, 1662, 1718, 1790

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

From Omar E. Pol, Mar 12 2011, Mar 15 2011, Mar 22 2011, Mar 25 2011: (Start)
We define an "I-toothpick" to consist of two connected toothpicks, as a bar of length 2. An I-toothpick with length 2 is formed by two toothpicks with length 1.
Note that in the physical model of the toothpick structure of A139250 the midpoint of a wooden toothpick of the new generation is superimposed on the endpoint of a wooden toothpick of the old generation. However, in the physical model of the I-toothpick structure the wooden toothpicks are not overlapping because all wooden toothpicks are connected by their endpoints.
a(n) is also the number of components after n-th stage in the toothpick structure of A139250, assuming the toothpicks have length 2.
Also, gullwing sequence starting from two opposite "gulls" (as a reflected gull in flight) such that the distance between their midpoints is equal to 2 (See A187220). The sequence gives the number of gulls in the structure after n-th stage.
Note that there is a correspondence between the gullwing structure and the I-toothpick structure, for example: a pair of opposite gulls in horizontal position in the gullwing structure is equivalent to a vertical I-toothpick with length 4 in the I-toothpick structure, such that the midpoint of each horizontal gull coincides with the midpoint of each vertical toothpick of the I-toothpick.
It appears this is also the connection between A147562 (the Ulam-Warburton cellular automaton) and the toothpick sequence A139250. The behavior of the function is similar to A147562 but here the structure is more complex. See Plot 2 button: A147562 vs A160164. See also A147562 vs A187220.
Also, B-toothpick sequence starting from two opposite "bells" such that the distance between their midpoints is equal to 4 (See A187220). We define a "B-toothpick" to consist of four arcs of length Pi/2 forming a "bell" similar to the Gauss function. A bell-shaped toothpick or B-toothpick or simply "bell" is formed by four Q-toothpicks (see A187210). A B-toothpick has length 2*Pi.  The sequence gives the number of bells in the structure after n stages.
We can see a correspondence between this structure and the I-toothpick structure of A139250. In this case, for example, a pair of opposite bells in horizontal position is equivalent to a vertical I-toothpick with length 8 in the I-toothpick structure, such that the midpoint of each horizontal bell coincides with the midpoint of each vertical toothpick of the I-toothpick.
Also, there is a fourth structure formed by isosceles right triangles, starting from two opposite triangles, since gulls or bells can be replaced by this type of triangles.
Note that the size of the toothpicks, gulls, bells and isosceles right triangles can be adjusted such that two or more of these structures can be overlaid.
(End)
The graph of this sequence is very close to the graphs of both A147562 and A169707 (see Plot 2). - Omar E. Pol, Feb 16 2015
It appears that a(n) is also the total number of ON cells after n-th stage in the half structure of the cellular automaton described in A169707 plus the total number of ON cells after n+1 stages in the half structure of the mentioned cellular automaton, without its central cell. See the illustration of the NW-NE-SE-SW version in A169707. - Omar E. Pol, Jul 26 2015
On the infinite Cairo pentagonal tiling consider the symmetric figure formed by two non-adjacent pentagons connected by a line segment joining two trivalent nodes. At stage 1 we start with one of these figures turned ON. a(n) is the number of ON cells in the structure after n-th stage, so a(1) = 2. The rule for the next stages is that the concave part of the figures of the new generation must be adjacent to the complementary convex part of the figures of the old generation. - Omar E. Pol, Mar 29 2018

			From _Omar E. Pol_, Aug 12 2013: (Start)
Illustration of initial terms:
.                                           _ _     _ _
.                     _ _ _ _   |_ _ _ _|    |_ _ _ _|
.       _ _   |_ _|    |_ _|    | |_ _| |   _|_|_ _|_|_
.   |    |    | | |    | | |      | | |        | | |
.   |   _|_   |_|_|    |_|_|      |_|_|     _ _|_|_|_ _
.             |   |   _|_ _|_   |_|_ _|_|    |_|_ _|_|
.                               |       |   _|_     _|_
.
.   2    6      14       22         30           46
.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..16384
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
Wikipedia, Cairo pentagonal tiling
Index entries for sequences related to cellular automata

Cf. A139250, A147562, A169707, A170903, A187210, A187220.

CoefficientList[Series[(2 x / ((1 - x) (1 + 2 x))) (1 + 2 x Product[1 + x^(2^k - 1) + 2 x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* Vincenzo Librandi, Feb 15 2015 *)

a(n) = 2*A139250(n).
a(n) = A187220(n+1) - 1. - Omar E. Pol, Mar 12 2011, Mar 22 2011
It appears that a(n) = A169707(n) + A170903(n), n >= 1. - Omar E. Pol, Feb 15 2015
It appears that a(n) = (A169707(n) - 1)/2 + (A169707(n+1) - 1)/2, n >= 1. - Omar E. Pol, Jul 24 2015

Zero inserted, more terms and edited by Omar E. Pol, Mar 12 2011

A160172 T-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468

Offset: 0

Omar E. Pol, Jun 01 2009

A T-toothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no T-toothpicks.
At round 1 we place a T-toothpick anywhere in the plane.
At round 2 we place three other T-toothpicks.
And so on...
The rule for adding a new T-toothpick is the following. A new T-toothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.
If the T-toothpick has three components then at the n-th round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every T-toothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section. - Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

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

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

a(n) = 2*A151920(n) + 2*A151920(n-1) + n + 1. - Charlie Neder, Feb 07 2019

Edited and extended by N. J. A. Sloane, Jan 01 2010

A147614 a(n) = number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 3, 7, 13, 19, 27, 39, 53, 63, 71, 83, 99, 119, 147, 183, 217, 235, 243, 255, 271, 291, 319, 355, 391, 419, 447, 487, 539, 607, 699, 803, 885, 919, 927, 939, 955, 975, 1003, 1039, 1075, 1103, 1131, 1171, 1223, 1291, 1383, 1487, 1571, 1615

Offset: 0

David Applegate, Apr 29 2009

a(n) is also the number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250. [From Omar E. Pol, May 18 2009]
It appears that the number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2*k, is equal to (2*k-2) * A139250(n) + a(n), k>0. See formulas in A160420 and A160422. [From Omar E. Pol, Nov 15 2010]
More generally, it appears that a(n) is also the number of grid points that are covered by the "special points" of the toothpicks of A139250, after n-th stage, assuming the toothpicks have length 2*k, k>0 and that each toothpick has three special points: the midpoint and two endpoints.
Note that if k>1 then also there are 2*k-2 grid points that are covered by each toothpick, but these points are not considered for this sequence. [From Omar E. Pol, Nov 15 2010]
Contribution from Omar E. Pol, Sep 16 2012 (Start):
It appears that a(n)/A139250(n) converge to 4/3.
It appears that a(n)/A160124(n) converge to 2.
It appears that a(n)/A139252(n) converge to 4.
(End)

David Applegate, Table of n, a(n) for n = 0..10135
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. A139250, A139251, A139252, A160124, A160420, A160422, A170884.

Since A160124(n) = 1+2*A139250(n)-A147614(n), n>0 (see A160124), and we have recurrences for A160125 (hence A160124) and A139250, we have a recurrence for this sequence. - N. J. A. Sloane, Feb 02 2010

a(n) = A187220(n+1)-A160124(n), n>0. - Omar E. Pol, Feb 15 2013

A152998 Toothpick sequence on the semi-infinite square grid.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 17, 21, 23, 27, 33, 39, 47, 61, 77, 85, 87, 91, 97, 103, 111, 125, 141, 151, 159, 173, 191, 211, 241, 285, 325, 341, 343, 347, 353, 359, 367, 381, 397, 407, 415, 429, 447, 467, 497, 541, 581, 599, 607, 621, 639

Offset: 0

Omar E. Pol, Dec 19 2008, Dec 23 2008, Jan 02 2008

nonn

Contribution from Omar E. Pol, Oct 01 2011 (Start):
On the semi-infinite square grid, at stage 0, we start from a vertical half toothpick at [(0,0),(0,1)]. This half toothpick represents one of the two components of the first toothpick placed in the toothpick structure of A139250. Consider only the toothpicks of length 2, so a(0) = 0.
At stage 1, we place an orthogonal toothpick of length 2 centered at the end, so a(1) = 1.
In each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
The sequence gives the number of toothpicks after n stages. A152968 (the first differences) gives the number of toothpicks added to the structure at n-th stage.
For more information see A139250. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..8191
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
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A152968.

Cf. A153000, A152978.

a(n) = (A139250(n+1)-1)/2.
From Omar E. Pol, Oct 01 2011: (Start)
a(n) = A139250(n+1) - A153003(n) + A153000(n-1) - 1, if n >= 1.
a(n) = A153003(n) - A153000(n-1), if n >= 1.
a(n) = 2*A153000(n-1) + 1, if n >= 1.
(End)
a(n) = (A187220(n+2) - 3)/4. - Omar E. Pol, Feb 18 2013

A187212 Q-toothpick sequence in the first quadrant.

Original entry on oeis.org

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

A267694 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,1). The endpoints of the left hand Q-toothpick are at (0,1) and (1,2). The endpoints of the right hand Q-toothpick are at (1,0) and (2,1). With a(0) = 0.

Original entry on oeis.org

0, 2, 5, 9, 16, 20, 27, 39, 54, 58, 65, 77, 92, 104, 127, 163, 194, 198, 205, 217, 232, 244, 267, 303, 334, 346

Offset: 0

Omar E. Pol, Jan 23 2016

a(n) is the total number of Q-toothpicks in the structure after n-th stage.
A267695 (the first differences) gives the number of Q-toothpicks added at n-th stage.
a(n) is also the total number of Q-toothpicks minus 1 after n+2 stages in the SW quadrant of the Q-toothpick structure of A187210, but with origing at (1,1) instead (0,0), and assuming that the initial Q-toothpick is in the first quadrant with origin at (0,0), if n>=1 . For more information see the comments and the formula of A187210.

Cf. A139250, A187210, A187220, A267695, A267698.

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.

A187216 Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.

Original entry on oeis.org

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

Omar E. Pol, Mar 30 2011

nonn

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.

			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.

Nathaniel Johnston, Table of n, a(n) for n = 0..200
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.]
Nathaniel Johnston, The Q-Toothpick Cellular Automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A160120, A160164, A187210, A187212, A187217, A187220.

a(15) - a(47) from Nathaniel Johnston, Apr 15 2011

A255049 a(n) = 2*A160552(n).

Original entry on oeis.org

0, 2, 2, 6, 2, 6, 10, 14, 2, 6, 10, 14, 10, 22, 34, 30, 2, 6, 10, 14, 10, 22, 34, 30, 10, 22, 34, 38, 42, 78, 98, 62, 2, 6, 10, 14, 10, 22, 34, 30, 10, 22, 34, 38, 42, 78, 98, 62, 10, 22, 34, 38, 42, 78, 98, 70, 42, 78, 106, 118, 162, 254, 258, 126, 2, 6, 10, 14, 10

Offset: 0

Omar E. Pol, Feb 13 2015

			Written as an irregular triangle in which row lengths are the terms of A011782 the sequence begins:
0;
2;
2,6;
2,6,10,14;
2,6,10,14,10,22,34,30;
2,6,10,14,10,22,34,30,10,22,34,38,42,78,98,62;
2,6,10,14,10,22,34,30,10,22,34,38,42,78,98,62,10,22,34,38,42,78,98,70,42,78,106,118,162,254,258,126;
It appears that row sums give 0 together with A004171, (see also A081294).
It appears that right border gives the nonnegative terms of A000918, (see also A095121).

Cf. A000918, A004171, A081294, A095121, A139250, A160164, A160552, A169707, A169708, A187220.

It appears that a(n) = A169708(n)/2, n >= 1.

Edited by Omar E. Pol, Feb 18 2015

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

A160164 Number of toothpicks after n-th stage in the I-toothpick structure of A139250.

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A160172 T-toothpick sequence (see Comments lines for definition).

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A147614 a(n) = number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2.

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A152998 Toothpick sequence on the semi-infinite square grid.

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A187212 Q-toothpick sequence in the first quadrant.

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A267694 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,1). The endpoints of the left hand Q-toothpick are at (0,1) and (1,2). The endpoints of the right hand Q-toothpick are at (1,0) and (2,1). With a(0) = 0.

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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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A187216 Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.

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A255049 a(n) = 2*A160552(n).

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