A019988 -id:A019988 - cp's OEIS frontend

A139250 Toothpick sequence (see Comments lines for definition).

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

A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 7, 5, 0, 1, 0, 1, 0, 1, 20, 16, 0, 1, 1, 0, 1, 0, 1, 60, 55, 0, 2, 1, 1, 0, 1, 0, 1, 204, 222, 0, 5, 2, 2, 1, 0, 1, 0, 1, 702, 950, 0, 12, 5, 5, 0, 1

Offset: 1

Pontus von Brömssen, Oct 22 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes (cells) with integer coordinates in D-dimensional space. For normal polyominoids, two cells are connected if they share a (d-1)-dimensional facet, but here we allow connections where the cells share a lower-dimensional face.
Each row is the counting sequence (by number of cells) of (D,d)-polyominoids with certain restrictions on the allowed connections between cells. Two cells have a connection of type (g,h) if they intersect in a (d-g)-dimensional unit cube and extend in d-h common dimensions. For example, d-dimensional polyominoes use connections of type (1,0), polyplets use connections of types (1,0) (edge connections) and (2,0) (corner connections), normal (3,2)-polyominoids use connections of types (1,0) ("soft" connections) and (1,1) ("hard" connections), hard polyominoids use connections of type (1,1).
Each row corresponds to a triple (D,d,C), where 1 <= d <= D and C is a set of pairs (g,h) with 1 <= g <= d and 0 <= h <= min(g, D-d). The k-th term of that row is the number of free k-celled (D,d)-polyominoids with connections of the types in C. Connections of types not in C are permitted, but the polyominoids must be connected through the specified connections only. For example, polyominoes may have cells that intersect in a point (g = 2) and hard polyominoids can have soft connections (h = 0) that are not needed to keep the polyominoids connected.
The rows are sorted first by D, then by d, and finally by a binary vector indicating which types of connections are allowed, where the connection types (g,h) are sorted lexicographically. (See table in cross-references.)
For each pair (D,d), the first row is 1, 0, 0, ..., corresponding to (D,d,{}) (no connections allowed).
The number of rows corresponding to given values of D and d is 2^((d+1)*(d+2)/2-1) if 2*d <= D and 2^((D-d+1)*(3*d-D+2)/2-1) otherwise.

			Array begins:
  n\k| 1  2  3  4  5   6    7     8      9     10      11       12
  ---+------------------------------------------------------------
   1 | 1  0  0  0  0   0    0     0      0      0       0        0
   2 | 1  1  1  1  1   1    1     1      1      1       1        1
   3 | 1  0  0  0  0   0    0     0      0      0       0        0
   4 | 1  1  1  1  1   1    1     1      1      1       1        1
   5 | 1  1  3  7 20  60  204   702   2526   9180   33989   126713
   6 | 1  2  5 16 55 222  950  4265  19591  91678  434005  2073783
   7 | 1  0  0  0  0   0    0     0      0      0       0        0
   8 | 1  1  2  5 12  35  108   369   1285   4655   17073    63600
   9 | 1  1  2  5 12  35  108   369   1285   4655   17073    63600
  10 | 1  2  5 22 94 524 3031 18770 118133 758381 4915652 32149296
  11 | 1  0  0  0  0   0    0     0      0      0       0        0
  12 | 1  1  1  1  1   1    1     1      1      1       1        1

Pontus von Brömssen, Table of n, a(n) for n = 1..210 (first 20 antidiagonals).
Pontus von Brömssen, Python programs that relate row numbers and parameter sets.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A366767 (fixed), A366768.
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C. Some sequences occur in more than one row. Notation used in the table:
  X: Allowed connection.
  -: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
  .: Not applicable for (D,d) in this row.
  !: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
  *: Whether a connection of type (g,h) is allowed or not is independent of h.
      |   |   | connections  |
      |   |   | g:1122233334 |
    n | D | d | h:0101201230 | sequence
  ----+---+---+--------------+---------
| 1 | 1 | * -......... | A063524
| 1 | 1 | * X......... | A000012
|!2 | 1 | * --........ | A063524
|!2 | 1 |   X-........ | A000012
| 2 | 1 |   -X........ | A361625
| 2 | 1 | * XX........ | A019988
| 2 | 2 | * -.-....... | A063524
| 2 | 2 | * X.-....... | A000105
| 2 | 2 | * -.X....... | A000105
| 2 | 2 | * X.X....... | A030222
|!3 | 1 | * --........ | A063524
|!3 | 1 |   X-........ | A000012
| 3 | 1 |   -X........ | A365654
| 3 | 1 | * XX........ | A365559
|!3 | 2 | * ----...... | A063524
|!3 | 2 |   X---...... | A000105
| 3 | 2 |   -X--...... | A365654
| 3 | 2 | * XX--...... | A075679
|!3 | 2 |   --X-...... | A000105
|!3 | 2 |   X-X-...... | A030222
| 3 | 2 |   -XX-...... | A365995
| 3 | 2 |   XXX-...... | A365997
| 3 | 2 |   ---X...... | A365999
| 3 | 2 |   X--X...... | A366001
| 3 | 2 |   -X-X...... | A366003
| 3 | 2 |   XX-X...... | A366005
| 3 | 2 | * --XX...... | A365652
| 3 | 2 |   X-XX...... | A366007
| 3 | 2 |   -XXX...... | A366009
| 3 | 2 | * XXXX...... | A365650
| 3 | 3 | * -.-..-.... | A063524
| 3 | 3 | * X.-..-.... | A038119
| 3 | 3 | * -.X..-.... | A038173
| 3 | 3 | * X.X..-.... | A268666
| 3 | 3 | * -.-..X.... | A038171
| 3 | 3 | * X.-..X.... | A363205
| 3 | 3 | * -.X..X.... | A363206
| 3 | 3 | * X.X..X.... | A272368
|!4 | 1 | * --........ | A063524
|!4 | 1 |   X-........ | A000012
| 4 | 1 |   -X........ | A366340
| 4 | 1 | * XX........ | A365561
|!4 | 2 | * -----..... | A063524
|!4 | 2 |   X----..... | A000105
| 4 | 2 |   -X---..... | A366338
| 4 | 2 | * XX---..... | A366334
|!4 | 2 |   --X--..... | A000105
|!4 | 2 |   X-X--..... | A030222
  ...
|!4 | 3 | * ----.--... | A063524
|!4 | 3 |   X---.--... | A038119
| 4 | 3 |   -X--.--... | A366340
| 4 | 3 | * XX--.--... | A366336
  ...
| 4 | 4 | * -.-..-...- | A063524
| 4 | 4 | * X.-..-...- | A068870
| 4 | 4 | * -.X..-...- | A365356
| 4 | 4 | * X.X..-...- | A365363
| 4 | 4 | * -.-..X...- | A365354
| 4 | 4 | * X.-..X...- | A365361
| 4 | 4 | * -.X..X...- | A365358
| 4 | 4 | * X.X..X...- | A365365
| 4 | 4 | * -.-..-...X | A365353
| 4 | 4 | * X.-..-...X | A365360
| 4 | 4 | * -.X..-...X | A365357
| 4 | 4 | * X.X..-...X | A365364
| 4 | 4 | * -.-..X...X | A365355
| 4 | 4 | * X.-..X...X | A365362
| 4 | 4 | * -.X..X...X | A365359
| 4 | 4 | * X.X..X...X | A365366
|!5 | 1 | * --........ | A063524
|!5 | 1 |   X-........ | A000012
| 5 | 1 |   -X........ |
| 5 | 1 | * XX........ | A365563

[3.6.3.6] refers to the face configuration of the rhombille tiling. - Peter Kagey, Mar 01 2020
If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - Joseph Myers, Jul 12 2020
These are also known as polytwigs, by association with their representation as polyedges. - Aaron N. Siegel, May 15 2022

Apparently the cells are circular blobs which must be connected diagonally and the polyominoes can be rotated by 90 degrees and turned over.
Also the number of essentially different (i.e., not related by reflections, translations or rotations) diagrams consisting of n nodes in Z^2 and n-1 horizontal or vertical edges of length 1 between pairs of nodes such that the resulting graph is connected (hence a tree). - Paul Boddington, Jul 27 2004
They are thus equivalent to a subset of the polyedges, counted by A019988, i.e., those that are treelike. - John Mason, Aug 20 2021
The number of treelike polyedges with n edges is a(n+1). - John Mason, Feb 12 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes with integer coordinates in D-dimensional space, where two cubes are connected if they share a (d-1)-dimensional facet. For example, (3,2)-polyominoids are normal polyominoids (A075679), (D,D)-polyominoids are D-dimensional polyominoes (A000105, A038119, A068870, ...), and (D,1)-polyominoids are polysticks in D dimensions (A019988, A365559, A365561, ...).

cp's OEIS Frontend

A139250 Toothpick sequence (see Comments lines for definition).

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A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).

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A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

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A096267 Number of fixed polyedges with n edges (number of ways of embedding connected undirected graphs with n edges into the plane square lattice, inequivalent up to translation).

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A365559 Number of free n-polysticks (or polyedges) in 3 dimensions.

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A056841 Number of diagonal polyominoes with n cells.

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A365561 Number of free n-polysticks (or polyedges) in 4 dimensions.

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A366334 Number of free (4,2)-polyominoids with n cells.

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A365563 Number of free n-polysticks (or polyedges) in 5 dimensions.

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