A161328 -id:A161328 - cp's OEIS frontend

A161329 First differences of A161328.

1, 3, 5, 7, 13, 11, 17, 15, 21, 23, 25, 27, 33, 27, 25, 15, 25, 35, 41, 55, 53, 59, 61, 59, 65, 63, 57, 47, 37, 47, 65, 71, 97, 95, 105, 95, 89, 83, 81, 87, 93, 79, 73, 79, 89, 107, 113, 119, 113, 115, 117, 135, 125, 127, 129, 135, 153, 135

Offset: 1

Omar E. Pol, Jun 07 2009

nonn

Number of E-Toothpicks added to the E-Toothpick structure at the n-th round.

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. A139251, A160121, A160173, A161328, A161331.

a(8) corrected and more terms added by R. J. Mathar, Jan 21 2010

A139250 Toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 79, 95, 123, 155, 171, 175, 183, 195, 207, 223, 251, 283, 303, 319, 347, 383, 423, 483, 571, 651, 683, 687, 695, 707, 719, 735, 763, 795, 815, 831, 859, 895, 935, 995, 1083, 1163, 1199, 1215, 1243, 1279, 1319, 1379

Offset: 0

Omar E. Pol, Apr 24 2008

A toothpick is a copy of the closed interval [-1,1]. (In the paper, we take it to be a copy of the unit interval [-1/2, 1/2].)
We start at stage 0 with no toothpicks.
At stage 1 we place a toothpick in the vertical direction, anywhere in the plane.
In general, given a configuration of toothpicks in the plane, at the next stage we add as many toothpicks as possible, subject to certain conditions:
- Each new toothpick must lie in the horizontal or vertical directions.
- Two toothpicks may never cross.
- Each new toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
The sequence gives the number of toothpicks after n stages. A139251 (the first differences) gives the number added at the n-th stage.
Call the endpoint of a toothpick "exposed" if it does not touch any other toothpick. The growth rule may be expressed as follows: at each stage, new toothpicks are placed so their midpoints touch every exposed endpoint.
This is equivalent to a two-dimensional cellular automaton. The animations show the fractal-like behavior.
After 2^k - 1 steps, there are 2^k exposed endpoints, all located on two lines perpendicular to the initial toothpick. At the next step, 2^k toothpicks are placed on these lines, leaving only 4 exposed endpoints, located at the corners of a square with side length 2^(k-1) times the length of a toothpick. - M. F. Hasler, Apr 14 2009 and others. For proof, see the Applegate-Pol-Sloane paper.
If the third condition in the definition is changed to "- Each new toothpick must have at exactly one of its endpoints touching the midpoint of an existing toothpick" then the same sequence is obtained. The configurations of toothpicks are of course different from those in the present sequence. But if we start with the configurations of the present sequence, rotate each toothpick a quarter-turn, and then rotate the whole configuration a quarter-turn, we obtain the other configuration.
If the third condition in the definition is changed to "- Each new toothpick must have at least one of its endpoints touching the midpoint of an existing toothpick" then the sequence n^2 - n + 1 is obtained, because there are no holes left in the grid.
A "toothpick" of length 2 can be regarded as a polyedge with 2 components, both on the same line. At stage n, the toothpick structure is a polyedge with 2*a(n) components.
Conjecture: Consider the rectangles in the sieve (including the squares). The area of each rectangle (A = b*c) and the edges (b and c) are powers of 2, but at least one of the edges (b or c) is <= 2.
In the toothpick structure, if n >> 1, we can see some patterns that look like "canals" and "diffraction patterns". For example, see the Applegate link "A139250: the movie version", then enter n=1008 and click "Update". See also "T-square (fractal)" in the Links section. - Omar E. Pol, May 19 2009, Oct 01 2011
From Benoit Jubin, May 20 2009: The web page "Gallery" of Chris Moore (see link) has some nice pictures that are somewhat similar to the pictures of the present sequence. What sequences do they correspond to?
For a connection to Sierpiński triangle and Gould's sequence A001316, see the leftist toothpick triangle A151566.
Eric Rowland comments on Mar 15 2010 that this toothpick structure can be represented as a 5-state CA on the square grid. On Mar 18 2010, David Applegate showed that three states are enough. See links.
Equals row sums of triangle A160570 starting with offset 1; equivalent to convolving A160552: (1, 1, 3, 1, 3, 5, 7, ...) with (1, 2, 2, 2, ...). Equals A160762: (1, 0, 2, -2, 2, 2, 2, -6, ...) convolved with 2*n - 1: (1, 3, 5, 7, ...). Starting with offset 1 equals A151548: [1, 3, 5, 7, 5, 11, 17, 15, ...] convolved with A078008 signed (A151575): [1, 0, 2, -2, 6, -10, 22, -42, 86, -170, 342, ...]. - Gary W. Adamson, May 19 2009, May 25 2009
For a three-dimensional version of the toothpick structure, see A160160. - Omar E. Pol, Dec 06 2009
From Omar E. Pol, May 20 2010: (Start)
Observation about the arrangement of rectangles:
It appears there is a nice pattern formed by distinct modular substructures: a central cross surrounded by asymmetrical crosses (or "hidden crosses") of distinct sizes and also by "nuclei" of crosses.
Conjectures: after 2^k stages, for k >= 2, and for m = 1 to k - 1, there are 4^(m-1) substructures of size s = k - m, where every substructure has 4*s rectangles. The total number of substructures is equal to (4^(k-1)-1)/3 = A002450(k-1). For example: If k = 5 (after 32 stages) we can see that:
a) There is a central cross, of size 4, with 16 rectangles.
b) There are four hidden crosses, of size 3, where every cross has 12 rectangles.
c) There are 16 hidden crosses, of size 2, where every cross has 8 rectangles.
d) There are 64 nuclei of crosses, of size 1, where every nucleus has 4 rectangles.
Hence the total number of substructures after 32 stages is equal to 85. Note that in every arm of every substructure, in the potential growth direction, the length of the rectangles are the powers of 2. (See illustrations in the links. See also A160124.) (End)
It appears that the number of grid points that are covered after n-th stage of the toothpick structure, assuming the toothpicks have length 2*k, is equal to (2*k-2)*a(n) + A147614(n), k > 0. See the formulas of A160420 and A160422. - Omar E. Pol, Nov 13 2010
Version "Gullwing": on the semi-infinite square grid, at stage 1, we place a horizontal "gull" with its vertices at [(-1, 2), (0, 1), (1, 2)]. At stage 2, we place two vertical gulls. At stage 3, we place four horizontal gulls. a(n) is also the number of gulls after n-th stage. For more information about the growth of gulls see A187220. - Omar E. Pol, Mar 10 2011
From Omar E. Pol, Mar 12 2011: (Start)
Version "I-toothpick": 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. The midpoint of an I-toothpick is touched by its two toothpicks. a(n) is also the number of I-toothpicks after n-th stage in the I-toothpick structure. The I-toothpick structure is essentially the original toothpick structure in which every toothpick is replaced by an I-toothpick. Note that in the physical model of the original toothpick structure 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. For the number of toothpicks in the I-toothpick structure see A160164 which also gives the number of gullwing in a gullwing structure because the gullwing structure of A160164 is equivalent to the I-toothpick structure. It also appears that the gullwing sequence A187220 is a supersequence of the original toothpick sequence A139250 (this sequence).
For the connection with the Ulam-Warburton cellular automaton see the Applegate-Pol-Sloane paper and see also A160164 and A187220.
(End)
A version in which the toothpicks are connected by their endpoints: on the semi-infinite square grid, at stage 1, we place a vertical toothpick of length 1 from (0, 0). At stage 2, we place two horizontal toothpicks from (0,1), and so on. The arrangement looks like half of the I-toothpick structure. a(n) is also the number of toothpicks after the n-th. - Omar E. Pol, Mar 13 2011
Version "Quarter-circle" (or Q-toothpick): a(n) is also the number of Q-toothpicks after the n-th stage in a Q-toothpick structure in the first quadrant. We start from (0,1) with the first Q-toothpick centered at (1, 1). The structure is asymmetric. For a similar structure but starting from (0, 0) see A187212. See A187210 and A187220 for more information. - Omar E. Pol, Mar 22 2011
Version "Tree": It appears that a(n) is also the number of toothpicks after the n-th stage in a toothpick structure constructed following a special rule: the toothpicks of the new generation have length 4 when they are placed on the infinite square grid (note that every toothpick has four components of length 1), but after every stage, one (or two) of the four components of every toothpick of the new generation is removed, if such component contains an endpoint of the toothpick and if such endpoint is touching the midpoint or the endpoint of another toothpick. The truncated endpoints of the toothpicks remain exposed forever. Note that there are three sizes of toothpicks in the structure: toothpicks of lengths 4, 3 and 2. A159795 gives the total number of components in the structure after the n-th stage. A153006 (the corner sequence of the original version) gives 1/4 of the total of components in the structure after the n-th stage. - Omar E. Pol, Oct 24 2011
From Omar E. Pol, Sep 16 2012: (Start)
It appears that a(n)/A147614(n) converges to 3/4.
It appears that a(n)/A160124(n) converges to 3/2.
It appears that a(n)/A139252(n) converges to 3.
Also:
It appears that A147614(n)/A160124(n) converges to 2.
It appears that A160124(n)/A139252(n) converges to 2.
It appears that A147614(n)/A139252(n) converges to 4.
(End)
It appears that a(n) is also the total number of ON cells after n-th stage in a quadrant of the structure of the cellular automaton described in A169707 plus the total number of ON cells after n+1 stages in a quadrant of the mentioned structure, without its central cell. See the illustration of the NW-NE-SE-SW version in A169707. See also the connection between A160164 and 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. 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. a(n) gives the number of figures that are ON in the structure after n-th stage. A160164(n) gives the number of ON cells in the structure after n-th stage. - Omar E. Pol, Mar 29 2018
From Omar E. Pol, Mar 06 2019: (Start)
The "word" of this sequence is "ab". For further information about the word of cellular automata see A296612.
Version "triangular grid": a(n) is also the total number of toothpicks of length 2 after n-th stage in the toothpick structure on the infinite triangular grid, if we use only two of the three axes. Otherwise, if we use the three axes, so we have the sequence A296510 which has word "abc".
The normal toothpick structure can be considered a superstructure of the Ulam-Warburton celular automaton since A147562(n) equals here the total number of "hidden crosses" after 4*n stages, including the central cross (beginning to count the crosses when their "nuclei" are totally formed with 4 quadrilaterals). Note that every quadrilateral in the structure belongs to a "hidden cross".
Also, the number of "hidden crosses" after n stages equals the total number of "flowers with six petals" after n-th stage in the structure of A323650, which appears to be a "missing link" between this sequence and A147562.
Note that the location of the "nuclei of the hidden crosses" is very similar (essentially the same) to the location of the "flowers with six petals" in the structure of A323650 and to the location of the "ON" cells in the version "one-step bishop" of the Ulam-Warburton cellular automaton of A147562. (End)
From Omar E. Pol, Nov 27 2020: (Start)
The simplest substructures are the arms of the hidden crosses. Each closed region (square or rectangle) of the structure belongs to one of these arms. The narrow arms have regions of area 1, 2, 4, 8, ... The broad arms have regions of area 2, 4, 8, 16 , ... Note that after 2^k stages, with k >= 3, the narrow arms of the main hidden crosses in each quadrant frame the size of the toothpick structure after 2^(k-1) stages.
Another kind of substructure could be called "bar chart" or "bar graph". This substructure is formed by the rectangles and squares of width 2 that are adjacent to any of the four sides of the toothpick structure after 2^k stages, with k >= 2. The height of these successive regions gives the first 2^(k-1) - 1 terms from A006519. For example: if k = 5 the respective heights after 32 stages are [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1]. The area of these successive regions gives the first 2^(k-1) - 1 terms of A171977. For example: if k = 5 the respective areas are [2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2].
For a connection to Mersenne primes (A000668) and perfect numbers (A000396) see A153006.
For a representation of the Wagstaff primes (A000979) using the toothpick structure see A194810.
For a connection to stained glass windows and a hidden curve see A336532. (End)
It appears that the graph of a(n) bears a striking resemblance to the cumulative distribution function F(x) for X the random variable taking values in [0,1], where the binary expansion of X is given by a sequence of independent coin tosses with probability 3/4 of being 1 at each bit. It appears that F(n/2^k)*(2^(2k+1)+1)/3 approaches a(n) for k large. - James Coe, Jan 10 2022

			a(10^10) = 52010594272060810683. - _David A. Corneth_, Mar 26 2015

D. 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
L. D. Pryor, The Inheritance of Inflorescence Characters in Eucalyptus, Proceedings of the Linnean Society of New South Wales, V. 79, (1954), p. 81, 83.
Richard P. Stanley, Enumerative Combinatorics, volume 1, second edition, chapter 1, exercise 95, figure 1.28, Cambridge University Press (2012), p. 120, 166.

N. J. A. Sloane, Table of n, a(n) for n = 0..65535
AlgoMotion, Toothpick Sequence Visualized with Circle of Fifths Harmony | 256 Steps, Youtube video (2024).
David Applegate, The movie version
David Applegate, Animation of first 32 stages
David Applegate, Animation of first 64 stages
David Applegate, Animation of first 128 stages
David Applegate, Animation of first 256 stages
David Applegate, C++ program to generate these animations - creates postscript for a specific n
David Applegate, Generates many postscripts, converts them to gifs, and glues the gifs together into an animation
David Applegate, Generates b-files for A139250, A139251, A147614
David Applegate, The b-files for A139250, A139251, A147614 side-by-side
David Applegate, A three-state CA for the toothpick structure
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.], which is also available at arXiv:1004.3036v2
Joe Champion, Ultimate toothpick pattern, Photo 1, Photo 2, Photo 3, Photo 4, Boise Math Circles, Boise State University. [Links updated by _P. Michael Hutchins_, Mar 03 2018]
Barry Cipra, What comes next?, Science (AAAS) 327: 943.
Steven R. Finch, Toothpicks and Live Cells, July 21, 2015. [Cached copy, with permission of the author]
Ulrich Gehmann, Martin Reiche, World mountain machine, Berlin, (2014), first edition, p. 205, 238, 253.
Mats Granvik, Additional illustration: Number blocks where each number tells how many times a point on the square grid is crossed or connected to by a toothpick, Jun 21 2009.
Gordon Hamilton, Three integer sequences from recreational mathematics, Video (2013?).
J. K. Hamilton, I. R. Hooper, and C. R. Lawrence, Exploring microwave absorption by non-periodic metasurfaces, Advanced Electromagnetics, 10(3), 1-6 (2021).
M. F. Hasler, Illustration of initial terms
M. F. Hasler, Illustrations (Three slides)
Brian Hayes, Joshua Trees and Toothpicks
Brian Hayes, Idealized Joshua tree, a figure from "Joshua Trees and Toothpicks" (see preceding link)
Brian Hayes, The Toothpick Sequence - Bit-Player
Benoit Jubin, Illustration of initial terms
Mathemaesthetics, 2'796'203 Toothpicks, 2'048 Generations, Youtube video (2021).
Ayliean McDonald, Toothpick fractal and breaking rules, Youtube video (2021)
Chris Moore, Gallery, see the section on David Griffeath's Cellular Automata.
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of initial terms using "gulls" (or G-toothpicks)
Omar E. Pol, Illustration of initial terms using quarter-circles (or Q-toothpicks)
Omar E. Pol, Illustration of the toothpick structure (after 23 steps)
Omar E. Pol, Illustration of patterns in the toothpick structure (after 32 steps)
Omar E. Pol, Illustration of patterns in the toothpick structure (after 32 steps) [Cached copy, with permission]
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (triangular grid and toothpick structure)
Omar E. Pol, Illustration of the substructures in the first quadrant (As pieces of a puzzle), after 32 stages
Omar E. Pol, Illustration of the potential growth direction of the arms of the substructures, after 32 stages
Olbaid Fractalium, Toothpick sequence Part 1, (Watch from minute 0:00 until 3:13), Youtube video (2023).
Programing Puzzles & Code Golf Stack Exchange, Generate toothpick sequence
L. D. Pryor, Illustration of initial terms (Fig. 2a)
L. D. Pryor, The Inheritance of Inflorescence Characters in Eucalyptus, Proceedings of the Linnean Society of New South Wales, V. 79, (1954), p. 79-89.
E. Rowland, Toothpick sequence from cellular automaton on square grid
E. Rowland, Initial stages of toothpick sequence from cellular automaton on square grid (includes Mathematica code)
K. Ryde, ToothpickTree.
Daniel Shiffman, Coding Challenge #126: Toothpicks, The Coding Train video (2018)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane and Brady Haran, Terrific Toothpick Patterns, Numberphile video (2018)
Alex van den Brandhof and Paul Levrie, Tandenstokerrij, Pythagoras, Wiskundetijdschrift voor Jongeren, 55ste Jaargang, Nummer 6, Juni 2016, (see the cover, pages 1, 18, 19 and the back cover).
Lu Wang, Minecraft Toothpicks N = 53
Wikipedia, Cairo pentagonal tiling
Wikipedia, H tree
Wikipedia, Toothpick sequence
Wikipedia, T-square (fractal)
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A000079, A002450, A006519, A139251, A139252, A139253, A147614, A139560, A152968, A152978, A152980, A152998, A153000, A153001, A153003, A153004, A153006, A153007, A000217, A007583, A007683, A000396, A000225, A000668, A006516, A006095, A019988, A160570, A160552, A000969, A001316, A151566, A160406, A160408, A160702, A078008, A151548, A001045, A147562, A160124, A160120, A160160, A160170, A160172, A161206, A161328, A161330, A171977, A194810, A296510, A296612, A299476, A299478, A323650, A336532.

G := (x/((1-x)*(1+2*x))) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k),k=0..20)); # N. J. A. Sloane, May 20 2009, Jun 05 2009
# From N. J. A. Sloane, Dec 25 2009: A139250 is T, A139251 is a.
a:=[0,1,2,4]; T:=[0,1,3,7]; M:=10;
for k from 1 to M do
a:=[op(a),2^(k+1)];
T:=[op(T),T[nops(T)]+a[nops(a)]];
for j from 1 to 2^(k+1)-1 do
a:=[op(a), 2*a[j+1]+a[j+2]];
T:=[op(T),T[nops(T)]+a[nops(a)]];
od: od: a; T;

CoefficientList[ Series[ (x/((1 - x)*(1 + 2x))) (1 + 2x*Product[1 + x^(2^k - 1) + 2*x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* Robert G. Wilson v, Dec 06 2010 *)
a[0] = 0; a[n_] := a[n] = Module[{m, k}, m = 2^(Length[IntegerDigits[n, 2]] - 1); k = (2m^2+1)/3; If[n == m, k, k + 2 a[n - m] + a[n - m + 1] - 1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 06 2018, after David A. Corneth *)

A139250(n,print_all=0)={my(p=[], /* set of "used" points. Points are written as complex numbers, c=x+iy. Toothpicks are of length 2 */
ee=[[0,1]], /* list of (exposed) endpoints. Exposed endpoints are listed as [c,d] where c=x+iy is the position of the endpoint, and d (unimodular) is the direction */
c,d,ne, cnt=1); print_all && print1("0,1"); n<2 && return(n);
for(i=2,n, p=setunion(p, Set(Mat(ee~)[,1])); /* add endpoints (discard directions) from last move to "used" points */
ne=[]; /* new (exposed) endpoints */
for( k=1, #ee, /* add endpoints of new toothpicks if not among the used points */
setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne,Set([[c,d]]));
setsearch(p, c-2*d) || ne=setunion(ne,Set([[c-2*d,-d]]));
); /* using Set() we have the points sorted, so it's easy to remove those which finally are not exposed because they touch a new toothpick */
forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] && k-- && ee=vecextract(ee,Str("^"k"..",k+1)));
cnt+=#ee; /* each exposed endpoint will give a new toothpick */
print_all && print1(","cnt));cnt} \\ M. F. Hasler, Apr 14 2009

\\works for n > 0
a(n) = {my(k = (2*msb(n)^2 + 1) / 3); if(n==msb(n),k , k + 2*a(n-msb(n)) + a(n - msb(n) + 1) - 1)}
msb(n)=my(t=0);while(n>>t>0,t++);2^(t-1)\\ David A. Corneth, Mar 26 2015

def msb(n):
    t = 0
    while n>>t > 0:
        t += 1
    return 2**(t - 1)
def a(n):
    k = (2 * msb(n)**2 + 1) / 3
    return 0 if n == 0 else k if n == msb(n) else k + 2*a(n - msb(n)) + a(n - msb(n) + 1) - 1
[a(n) for n in range(101)]  # Indranil Ghosh, Jul 01 2017, after David A. Corneth's PARI script

a(2^k) = A007583(k), if k >= 0.
a(2^k-1) = A006095(k+1), if k >= 1.
a(A000225(k)) - a((A000225(k)-1)/2) = A006516(k), if k >= 1.
a(A000668(k)) - a((A000668(k)-1)/2) = A000396(k), if k >= 1.
G.f.: (x/((1-x)*(1+2*x))) * (1 + 2*x*Product_{k>=0} (1 + x^(2^k-1) + 2*x^(2^k))). - N. J. A. Sloane, May 20 2009, Jun 05 2009
One can show that lim sup a(n)/n^2 = 2/3, and it appears that lim inf a(n)/n^2 is 0.451... - Benoit Jubin, Apr 15 2009 and Jan 29 2010, N. J. A. Sloane, Jan 29 2010
Observation: a(n) == 3 (mod 4) for n >= 2. - Jaume Oliver Lafont, Feb 05 2009
a(2^k-1) = A000969(2^k-2), if k >= 1. - Omar E. Pol, Feb 13 2010
It appears that a(n) = (A187220(n+1) - 1)/2. - Omar E. Pol, Mar 08 2011
a(n) = 4*A153000(n-2) + 3, if n >= 2. - Omar E. Pol, Oct 01 2011
It appears that a(n) = A160552(n) + (A169707(n) - 1)/2, n >= 1. - Omar E. Pol, Feb 15 2015
It appears that a(n) = A255747(n) + A255747(n-1), n >= 1. - Omar E. Pol, Mar 16 2015
Let n = msb(n) + j where msb(n) = A053644(n) and let a(0) = 0. Then a(n) = (2 * msb(n)^2 + 1)/3 + 2 * a(j) + a(j+1) - 1. - David A. Corneth, Mar 26 2015
It appears that a(n) = (A169707(n) - 1)/4 + (A169707(n+1) - 1)/4, n >= 1. - Omar E. Pol, Jul 24 2015

Verified and extended, a(49)-a(53), using the given PARI code by M. F. Hasler, Apr 14 2009
Edited by N. J. A. Sloane, Apr 29 2009, incorporating comments from Omar E. Pol, M. F. Hasler, Rob Pratt, Jaume Oliver Lafont, Franklin T. Adams-Watters, R. J. Mathar, David W. Wilson, David Applegate, Benoit Jubin and others.
Further edited by N. J. A. Sloane, Jan 28 2010

A160120 Y-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 67, 76, 85, 106, 121, 142, 169, 220, 247, 256, 265, 286, 301, 322, 349, 400, 433, 454, 481, 532, 583, 640, 709, 826, 907, 928, 937, 958, 973, 994, 1021, 1072, 1105, 1126, 1153, 1204, 1255, 1312, 1381, 1498, 1585, 1618, 1645

Offset: 0

Omar E. Pol, May 02 2009

nonn

A Y-toothpick (or Y-shaped toothpick) is formed from three toothpicks of length 1, like a star with three endpoints and only one middle-point.
On the infinite triangular grid, we start at round 0 with no Y-toothpicks.
At round 1 we place a Y-toothpick anywhere in the plane.
At round 2 we add three more Y-toothpicks. After round 2, in the structure there are three rhombuses and a hexagon.
At round 3 we add three more Y-toothpicks.
And so on ... (see illustrations).
The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.
The Y-toothpick pattern has a recursive, fractal (or fractal-like) structure.
Note that, on the infinite triangular grid, a Y-toothpick can be represented as a polyedge with three components. In this case, at the n-th round, the structure is a polyedge with 3*a(n) components.
This structure is more complex than the toothpick structure of A139250. For example, at some rounds we can see inward growth.
The structure contains distinct polygons which have side length equal to 1.
Observation: It appears that the region of the structure where all grid points are covered is formed only by three distinct polygons:
- Triangles
- Rhombuses
- Concave-convex hexagons
Holes in the structure: Also, we can see distinct concave-convex polygons which contains a region where there are no grid points that are covered, for example:
- Decagons   (with  1 non-covered grid point)
- Dodecagons (with  4 non-covered grid points)
- 18-gons    (with  7 non-covered grid points)
- 30-gons    (with 26 non-covered grid points)
- ...
Observation: It appears that the number of distinct polygons that contain non-covered grid points is infinite.
This sequence appears to be related to powers of 2. For example:
Conjecture: It appears that if n = 2^k, k>0, then, between the other polygons, there appears a new centered hexagon formed by three rhombuses with side length = 2^k/2 = n/2.
Conjecture: Consider the perimeter of the structure. It appears that if n = 2^k, k>0, then the structure is a triangle-shaped polygon with A000225(k)*6 sides and a half toothpick in each vertice of the "triangle".
Conjecture: It appears that if n = 2^k, k>0, then the ratio of areas between the Y-toothpick structure and the unitary triangle is equal to A006516(k)*6.
See the entry A139250 for more information about the growth of "standard" toothpicks.
See also A160715 for another version of this structure but without internal growth of Y-toothpicks. [Omar E. Pol, May 31 2010]
For an alternative visualization replace every single toothpick with a rhombus, or in other words, replace every Y-toothpick with the "three-diamond" symbol, so we have a cellular automaton in which a(n) gives the total number of "three-diamond" symbols after n-th stage and A160167(n) counts the total number of "ON" diamonds in the structure after n-th stage. See also A253770. - Omar E. Pol, Dec 24 2015
The behavior is similar to A153006 (see the graph). - Omar E. Pol, Apr 03 2018

JungHwan Min, Table of n, a(n) for n = 0..5000
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.]
David Applegate, The movie version
Ayliean McDonald, Y-toothpick fractal, Twitter video (2021).
Olbaid Fractalium, Toothpick sequence Part 2, Youtube video (2023).
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the structure after 17 stages
Omar E. Pol, Illustration of the structure after 34 stages, from Applegate's movie version.
Omar E. Pol, Illustration: Fractal recursion, general step. (1)
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Toothpick sequence: A139250.

Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.

YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 0 && Count[out, 2] >= 2, 1, in == 0 && Count[out, 2] == 1, 2, True, in]]; A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)
A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{435225738745686506433286166261571728070, 3, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)

More terms from David Applegate, Jun 14 2009, Jun 18 2009

A161206 V-toothpick (or honeycomb) sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 31, 43, 57, 69, 81, 99, 123, 153, 183, 211, 241, 261, 273, 291, 317, 351, 393, 443, 499, 553, 597, 645, 709, 791, 871, 939, 1005, 1041, 1053, 1071, 1097, 1131, 1173, 1223, 1281, 1339, 1393, 1459, 1549, 1663, 1789, 1911, 2031, 2133, 2193

Offset: 0

Omar E. Pol, Jun 08 2009

nonn

A V-toothpick is constructed from two toothpicks of length 1 with a 120-degree angle between them, forming a V.
On the infinite hexagonal grid, we start at round 0 with no V-toothpicks.
At round 1 we place a V-toothpick anywhere in the plane.
At round 2 we place two other V-toothpicks. Note that, after round 2, in the structure there are three V-toothpicks, with seven 120-degree angles and one 240-degree angle.
At round 3 we place four other V-toothpicks.
And so on...
The structure looks like an unfinished honeycomb.
The sequence gives the number of V-toothpicks after n rounds. A161207 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the growth of toothpicks.
Note that, on the infinite hexagonal grid, a V-toothpick can be represented as a polyedge with two components. In this case, at n-th round, the structure is a polyedge with 2*a(n) components (or 2*a(n) toothpicks).
In the structure we can see distinct closed polygonal regions with side length equal to 1, for example: regular hexagons, concave decagons, concave dodecagons.

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.]
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A160120, A160172, A161207, A161328, A161330.

Terms beyond a(19) from R. J. Mathar, Jan 21 2010

A161330 Snowflake (or E-toothpick) sequence (see Comments lines for definition).

Original entry on oeis.org

0, 2, 8, 14, 20, 38, 44, 62, 80, 98, 128, 146, 176, 218, 224, 242, 260, 290, 344, 374, 452, 494, 548, 626, 668, 734, 812, 830, 872, 914, 968, 1058, 1124, 1250, 1340, 1430, 1532, 1598, 1676, 1766, 1856, 1946, 2000, 2066, 2180, 2258, 2384, 2510, 2612, 2714, 2852, 2954, 3116, 3218, 3332, 3494, 3620, 3782, 3896, 3998, 4100

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

This sequence is an E-toothpick sequence (cf. A161328) but starting with two back-to-back E-toothpicks.
On the infinite triangular grid, we start at round 0 with no E-toothpicks.
At round 1 we place two back-to-back E-toothpicks, forming a star with six endpoints.
At round 2 we add six more E-toothpicks.
At round 3 we add six more E-toothpicks.
And so on ... (see the illustrations).
The rule for adding new E-toothpicks is as follows. Each E has three ends, which initially are free. If the ends of two E's meet, those ends are no longer free. To go from round n to round n+1, we add an E-toothpick at each free end (extending that end in the direction it is pointing), subject to the condition that no end of any new E can touch any end of an existing E from round n or earlier. (Two new E's are allowed to touch.)
The sequence gives the number of E-toothpicks in the structure after n rounds. A161331 (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.
Note that, on the infinite triangular grid, a E-toothpick can be represented as a polyedge with three components. In this case, at n-th round, the structure is a polyedge with 3*a(n) components.

David Applegate, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
David Applegate, Illustration of structure after 32 stages. (Contains 1124 E-toothpicks.)
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.]
Ed Jeffery, Illustration of A161330 structure after 32 stages, with E-toothpicks replace by rhombi (the figure on the right is the complementary structure)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks) [From _Omar E. Pol_, Dec 06 2009]
N. J. A. Sloane, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160120, A160172, A161206, A161328, A161331, A161333.

[No formula or recurrence is known, - N. J. A. Sloane, Oct 13 2023]

For n >= 2, a(n) = 2 + Sum_{k=2..n} 6*A220498(k-1) - 6. - Christopher Hohl, Feb 24 2019. [This is a restatement of the definition. - N. J. A. Sloane, Oct 13 2023]

a(9)-a(12) from N. J. A. Sloane, Dec 07 2012

Corrected and extended by David Applegate, Dec 12 2012

A172310 L-toothpick sequence (see Comment lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 33, 47, 61, 79, 97, 117, 141, 165, 203, 237, 279, 313, 339, 367, 399, 437, 489, 543, 607, 665, 733, 793, 853, 903, 969, 1039, 1109, 1183, 1233, 1285, 1345, 1399, 1463, 1529, 1613, 1701, 1817, 1923, 2055, 2155, 2291, 2417, 2557, 2663, 2781, 2881, 3003, 3109, 3247, 3361, 3499, 3631, 3783, 3939

Offset: 0

Omar E. Pol, Jan 31 2010

nonn

We define an "L-toothpick" to consist of two line segments forming an "L".
There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).
The rule for the n-th stage:
If n is odd then we add the large L-toothpicks to the structure, otherwise we add the small L-toothpicks to the structure.
Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.
The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation.
We start at stage 0 with no L-toothpicks.
At stage 1 we place a large L-toothpick in the horizontal direction, as a "V", anywhere in the plane (Note that there are two exposed endpoints).
At stage 2 we place two small L-toothpicks.
At stage 3 we place four large L-toothpicks.
At stage 4 we place six small L-toothpicks.
And so on...
The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th stage.
For more information see A139250, the toothpick sequence.
In calculating the extension, the "special rule" was strengthened to prohibit intersections as well as overlappings. [From John W. Layman, Feb 04 2010]
Note that the endpoints of the L-toothpicks of the new generation can touch the L-toothpìcks of old generations but the crosses and overlaps are prohibited. - Omar E. Pol, Mar 26 2016
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A194270). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

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
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

For a similar version see A172304.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172311, A172312, A194270, A220500.

Terms a(9)-a(41) from John W. Layman, Feb 04 2010

Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

A161331 First differences of A161330.

Original entry on oeis.org

0, 2, 6, 6, 6, 18, 6, 18, 18, 18, 30, 18, 30, 42, 6, 18, 18, 30, 54, 30, 78, 42, 54, 78, 42, 66, 78, 18, 42, 42, 54, 90, 66, 126, 90, 90, 102, 66, 78, 90, 90, 90, 54, 66, 114, 78, 126, 126, 102, 102, 138, 102, 162, 102, 114, 162, 126, 162, 114, 102, 102, 126, 186, 186, 150, 138, 126, 162, 162, 186, 198, 114, 114, 162

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

Number of E-toothpicks added to the snowflake structure at n-th round.

David Applegate, Table of n, a(n) for n = 0..1000
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, A160121, A160173, A161328, A161329, A161330.

More terms from David Applegate, Dec 13 2012

A161336 Snowflake tree sequence: (A161330(n+1) - 2)/6.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 13, 16, 21, 24, 29, 36, 37, 40, 43, 48, 57, 62, 75, 82, 91, 104, 111, 122, 135, 138, 145, 152, 161, 176, 187, 208, 223, 238, 255, 266, 279, 294, 309, 324, 333, 344, 363, 376, 397, 418, 435, 452, 475, 492, 519, 536, 555, 582, 603, 630, 649, 666, 683

Offset: 0

Omar E. Pol, Jun 09 2009

nonn

This is an E-toothpick sequence. On a triangular graph paper consider an infinite 60-degree wedge in which there is a single (and virtual) toothpick connected to its vertex. At stage 0 we start with no E-toothpicks. At stage 1 we place an E-toothpick, and so on. The sequence gives the number of E-toothpicks in the structure after n stages. A211974 (the first differences) gives the number added at the n-th stage. The structure is the tree that arise from one of the six spokes of the structure of A213360 which is essentially the same as the E-toothpick (or snowflake) structure of A161330. For n >> 1 the structure looks like a quadrilateral formed by two scalene right triangles which are joined at their hypotenuses. - Omar E. Pol, Dec 19 2012

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, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A161328, A161330, A161337, A161338, A211974, A213360.

a(n) = A213360(n)/6. - Omar E. Pol, Dec 20 2012

Extended and edited by Omar E. Pol, Dec 19 2012

A172304 L-toothpick sequence starting with two opposite L-toothpicks.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 46, 62, 70, 86, 110, 134, 166, 190, 238, 278, 302, 318, 342, 382, 430, 470, 526, 582, 646, 710, 782, 838, 902, 950, 1030, 1118, 1150, 1182, 1246, 1318, 1382, 1422, 1486, 1566, 1662, 1766, 1910, 2006, 2134, 2254, 2414, 2526, 2622

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

The same as A172310 but starting with two L-toothpicks.
We start at stage 0 with no L-toothpicks.
At stage 1 we place two large L-toothpicks in the horizontal direction, as a "X", anywhere in the plane.
At stage 2 we place four small L-toothpicks.
At stage 3 we add eight more large L-toothpicks.
At stage 4 we add eight more small L-toothpicks.
And so on ...
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A212008). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

Yan Sheng Ang, Table of n, a(n) for n = 0..201
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

For a similar version see A172310.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172305, A172306, A172307, A172308, A172309, A172311, A172312, A172313, A212008, A220500.

Terms beyond a(14) from Yan Sheng Ang, Dec 10 2012

A172308 L-toothpick sequence in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 17, 21, 27, 33, 41, 47, 59, 69, 75, 79, 85, 95, 107, 117, 131, 145, 161, 177, 195, 209, 225, 237, 257, 279, 287, 295, 311, 329, 345, 355, 371, 391, 415, 441, 477, 501, 533, 563, 603, 631, 655

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

The same as A172310 and A172304, but starting from half L-toothpick in the first quadrant.
Note that if n is odd then we add the small L-toothpicks to the structure, otherwise we add the large L-toothpicks to the structure.
We start at stage 0 with half L-toothpick: A segment from (0,0) to (1,1).
At stage 1 we place a small L-toothpick at the exposed toothpick end.
At stage 2 we place two large L-toothpicks.
At stage 3 we place two small L-toothpicks.
At stage 4 we place two large L-toothpicks.
And so on...
The sequence gives the number of L-toothpicks after n stages. A172309 (the first differences) gives the number of L-toothpicks added at the n-th stage.

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, A153000, A160120, A160170, A160172, A161206, A161328, A172304, A172305, A172306, A172307, A172309, A172310, A172311, A172312, A172313.

a(17)-a(47) from Robert Price, Jun 17 2019

cp's OEIS Frontend

A161329 First differences of A161328.

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A139250 Toothpick sequence (see Comments lines for definition).

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A160120 Y-toothpick sequence (see Comments lines for definition).

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A161206 V-toothpick (or honeycomb) sequence (see Comments lines for definition).

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A161330 Snowflake (or E-toothpick) sequence (see Comments lines for definition).

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A172310 L-toothpick sequence (see Comment lines for definition).

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A161331 First differences of A161330.

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A161336 Snowflake tree sequence: (A161330(n+1) - 2)/6.

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A172304 L-toothpick sequence starting with two opposite L-toothpicks.

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