A147562 -id:A147562 - cp's OEIS frontend

A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

1, 5, 9, 21, 25, 37, 53, 85, 89, 101, 117, 149, 165, 201, 261, 341, 345, 357, 373, 405, 421, 457, 517, 597, 613, 649, 709, 793, 853, 965, 1173, 1365, 1369, 1381, 1397, 1429, 1445, 1481, 1541, 1621, 1637, 1673, 1733, 1817, 1877, 1989, 2197, 2389, 2405, 2441, 2501

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

Partial sums of A162793.
Also, total number of ON cells at stage n of the two-dimensional cellular automaton defined as follows: replace every "vertical" toothpick of length 2 with a centered unit square "ON" cell, so we have a cellular automaton which is similar to both A147562 and A169707 (this is the "one-step bishop" version). For the "one-step rook" version we use toothpicks of length sqrt(2), then rotate the structure 45 degrees and then replace every toothpick with a unit square "ON" cell. For the illustration of the sequence as a cellular automaton we now have three versions: the original version with toothpicks, the one-step rook version and one-step bishop version. Note that the last two versions refer to the standard ON cells in the same way as the two versions of A147562 and the two versions of A169707. It appears that the graph of this sequence lies between the graphs of A147562 and A169707. Also, it appears that this sequence shares infinitely many terms with both A147562 and A169707, see Formula section and Example section. - Omar E. Pol, Feb 20 2015
It appears that this is also a bisection (the odd terms) of A255747.

			From _Omar E. Pol_, Feb 18 2015: (Start)
Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
    1;
    5;
    9, 21;
   25, 37, 53, 85;
   89,101,117,149,165,201,261,341;
  345,357,373,405,421,457,517,597,613,649,709,793,853,965,1173,1365;
  ...
The right border gives the positive terms of A002450.
(End)
It appears that T(j,k) = A147562(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements of the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015

Michel Marcus, Table of n, a(n) for n = 1..10000
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. A002450, A048645, A139250, A139251, A147562, A153000, A159791, A159792, A160164, A160552, A162793, A162794, A162796, A162797, A169707, A255263, A255264, A255747.

It appears that a(n) = A147562(n) = A169707(n), if n is a term of A048645, otherwise A147562(n) < a(n) < A169707(n). - Omar E. Pol, Feb 20 2015
It appears that a(n) = (A169707(2n) - 1)/4 = A255747(2n-1). - Omar E. Pol, Mar 07 2015
a(n) = 1 + 4*A255737(n-1). - Omar E. Pol, Mar 08 2015

More terms from N. J. A. Sloane, Dec 28 2009

A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 4, 16, 28, 64, 76, 112, 148, 256, 268, 304, 340, 448, 484, 592, 700, 1024, 1036, 1072, 1108, 1216, 1252, 1360, 1468, 1792, 1828, 1936, 2044, 2368, 2476, 2800, 3124, 4096, 4108, 4144, 4180, 4288, 4324, 4432, 4540, 4864, 4900, 5008, 5116, 5440, 5548, 5872, 6196

Offset: 0

Omar E. Pol, May 20 2009

On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.
At round 1, we turn ON four cells, forming a square.
The rule for n > 1: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
Therefore:
At Round 2, we turn ON twelve cells around the square.
At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.
And so on.
For the first differences see the entry A161411.
Shows a fractal behavior similar to the toothpick sequence A139250.
A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4.
When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).
a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009
From Omar E. Pol, Mar 28 2011: (Start)
Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.
Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.)
The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage.
(End)

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
    4;
   16;
   28,  64;
   76, 112, 148, 256;
  268, 304, 340, 448, 484, 592, 700, 1024;
  ...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A256534 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                         _ _ _ _
.      |  _ _  |                       |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _     _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  |   |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_|   |_|  _ _  |_| |
.          |   | |  _|_|_ _ _ _|_|_  | |   |
.          |  _| |_|  _ _     _ _  |_| |_  |
.          | | |_ _| |  _|_ _|_  | |_ _| | |
.          | |_ _| | |_|  _ _  |_| | |_ _| |
.          |       |   | |   | |   |       |
.          |  _ _  |  _| |_ _| |_  |  _ _  |
.          | |  _|_| | |_ _ _ _| | |_|_  | |
.          | |_|  _| |_ _|   |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_     _| |_ _| |_  |
.       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _|   |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                       | |_ _| |
.      |_ _ _ _|                       |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.],
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
Omar E. Pol, Illustration of initial terms (2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.

RasterGraphics[state_?MatrixQ,colors_Integer:2,opts___]:=
Graphics[Raster[Reverse[1-state/(colors -1)]],
AspectRatio ->(AspectRatio/.{opts}/.AspectRatio ->Automatic),
Frame ->True, FrameTicks ->None, GridLines ->None];
rule=1340761804646523638425234105559798690663900360577570370705802859623\
705267234688669629039040624964794287326910250673678735142700520276191850\
5902735959769690
Show[GraphicsArray[Map[RasterGraphics,CellularAutomaton[{rule, {2,
{{4,2,1}, {32,16,8}, {256,128,64}}}, {1,1}}, {{{1,1}, {1,1}}, 0}, 9,-10]]]];
ca=CellularAutomaton[{rule,{2,{{4,2,1},{32,16,8},{256,128,64}}},{1,
1}},{{{1,1},{1,1}},0},99,-100];
Table[Total[ca[[i]],2],{i,1,Length[ca]}]
(* John W. Layman, Sep 01 2009; Sep 02 2009 *)
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

A160410(n)=sum(i=0,n-1,3^norml2(binary(i)))<<2 \\ M. F. Hasler, Dec 04 2012

Equals 4*A130665. This provides an explicit formula for a(n). - N. J. A. Sloane, Jul 13 2009

a(2^k) = (2*(2^k))^2 for k>=0.

Edited by David Applegate and N. J. A. Sloane, Jul 13 2009

A187220 Gullwing sequence (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 15, 23, 31, 47, 71, 87, 95, 111, 135, 159, 191, 247, 311, 343, 351, 367, 391, 415, 447, 503, 567, 607, 639, 695, 767, 847, 967, 1143, 1303, 1367, 1375, 1391, 1415, 1439, 1471, 1527, 1591, 1631, 1663, 1719, 1791, 1871, 1991, 2167, 2327, 2399, 2431

Offset: 0

Omar E. Pol, Mar 07 2011

The Gullwing (or G-toothpick) sequence is a special type of toothpick sequence. It appears that this is a superstructure of A139250.
We define a "G-toothpick" to consist of two arcs of length Pi/2 forming a "gullwing" whose total length is equal to Pi = 3.141592...
A gullwing-shaped toothpick or G-toothpick or simply "gull" is formed by two Q-toothpicks (see A187210).
A G-toothpick has a midpoint and two endpoints. An endpoint is said to be "exposed" if it is not the midpoint or endpoint of any other G-toothpick.
The sequence gives the number of G-toothpicks in the structure after n stages. A187221 (the first differences) gives the number of G-toothpicks added at n-th stage.
a(n) is also the diameter of a circle whose circumference equals the total length of all gulls in the gullwing structure after n stages.
It appears that the gullwing pattern has a recursive, fractal-like structure. The animation shows the fractal-like behavior.
Note that the structure contains many different types of geometrical figures, for example: circles, hearts, etc. All figures are formed by arcs.
It appears that there are infinitely many types of circular shapes, which are related to the rectangles of the toothpick structure of A139250.
It also appears that the structure contains a nice pattern formed by distinct modular substructures: one central cross surrounded by several asymmetrical crosses (or "hidden crosses") of distinct sizes, and also several "nuclei" of crosses. This pattern is essentially similar to the crosses of A139250 but here the structure is harder to see. For example, consider the nucleus of a cross; in the toothpick structure a nucleus is formed by two squares and two rectangles but here a nucleus is formed by two circles and two hearts.
It appears furthermore that this structure has connections with the square-cross fractal and with the T-square fractal, just as in the case of the toothpick structure of A139250.
For more information see A139250 and A187210.
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 A187220). - Omar E. Pol, Mar 11 2011, Mar 13 2011
From Omar E. Pol, Mar 25 2011: (Start)
If we remove the first gull of the structure so we can see that there is a correspondence between the gullwing structure and the I-toothpick structure of A139250, 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. See A160164.
Also, B-toothpick sequence. 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 B-toothpicks in the structure after n stages.
Also, if we remove the first bell of the structure, we can find 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. See A160164.
Also, for this sequence there is a third structure formed by isosceles right triangles since gulls or bells can be replaced by these triangles.
Note that the size of the gulls, bells and triangles can be adjusted such that two or three of these structures can be overlaid.
(End)
Also, it appears that if we let k=floor(log_2(n)), then for n >= 1, a(2^k) = (4^(k+1) + 5)/3 - 2^(k+1).  Otherwise, a(n)=(4^(k+1) + 5)/3 + 8*A153006(n-1-2^k). - Christopher Hohl, Dec 19 2018

			On the infinite square grid we start at stage 0 with no G-toothpicks, so a(0) = 0.
At stage 1 we place a G-toothpick:
Midpoint : (0,-1)
Endpoints: (-1,0) and (1,0)
So a(1) = 1. There are two exposed endpoints.
At stage 2 we place two G-toothpicks:
Midpoint of the left G-toothpick : (-1,0)
Endpoints of the left G-toothpick: (-2,1) and (-2,-1)
Midpoint of the right G-toothpick : (1,0)
Endpoints of the right G-toothpick: (2,1) and (2,-1)
So a(2) = 1+2 = 3. There are four exposed endpoints.
And so on...

Iain Fox, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
David Applegate, Illustration for a(16) = 311 [Created from the movie version, see previous link]
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.]
Brady Haran and N. J. A. Sloane, Terrific Toothpick Patterns, Numberphile video (2018).
Omar E. Pol, Illustration of initial terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A002450, A139250, A147562, A160164, A187210, A187221, A187222.

Join[{0, 1}, Rest[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] + 1 ]] (* Vincenzo Librandi, Jul 02 2017 *)

A139250(n) = my(msb(m) = 2^(#binary(m)-1), k = (2*msb(n)^2 + 1) / 3); if(n==msb(n), k , k + 2*A139250(n-msb(n)) + A139250(n - msb(n) + 1) - 1)
a(n) = if(n<2, n, 1 + 2*A139250(n-1)) \\ Iain Fox, Dec 10 2018

def msb(n):
    t=0
    while n>>t>0: t+=1
    return 2**(t - 1)
def a139250(n):
    k=(2*msb(n)**2 + 1)//3
    return 0 if n==0 else k if n==msb(n) else k + 2*a139250(n - msb(n)) + a139250(n - msb(n) + 1) - 1
def a(n): return 0 if n==0 else 1 + 2*a139250(n - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 01 2017

a(n) = 1 + 2*A139250(n-1), for n >= 1.
a(n) = 1 + A160164(n-1), for n >= 1. - [Suggested by Omar E. Pol, Mar 13 2011, proved by Nathaniel Johnston, Mar 22 2011]
The formula involving A160164 can be seen by identifying a Gullwing in the n-th generation (n >= 2) with midpoint at (x,y) and endpoints at (x-1,y+1) and (x+1,y+1) with a toothpick in the (n-1)st generation with endpoints at (x,y-1) and (x,y+1) -- this toothpick from (x,y-1) to (x,y+1) should be considered as having length ONE (i.e., it is HALF of an I-toothpick). The formula involving A139250 follows as a result of the relationship between A139250 and A160164.
a(n) = A147614(n-1) + A160124(n-1), n >= 2. - Omar E. Pol, Feb 15 2013
a(n) = 7 + 8*A153000(n-3), n >= 3. - Omar E. Pol, Feb 16 2013

A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 12, 19, 20, 23, 26, 33, 36, 43, 50, 65, 66, 69, 72, 79, 82, 89, 96, 111, 114, 121, 128, 143, 150, 165, 180, 211, 212, 215, 218, 225, 228, 235, 242, 257, 260, 267, 274, 289, 296, 311, 326, 357, 360, 367, 374, 389, 396, 411, 426, 457, 464, 479, 494, 525, 540, 571, 602, 665, 666, 669, 672, 679, 682, 689

Offset: 0

Omar E. Pol, Jan 19 2016

nonn

Conjecture: this is also the "tree" sequence in a 120-degree sector of the cellular automaton of A266532.
It appears that this is also the partial sums of A038573.
a(n) is also the total number of ON cells after n-th stage in the tree that arises from one of the four spokes in a 90-degree sector of the cellular automaton A160720 on the square grid.
Note that the structure of A160720 is also the "outward" version of the Ulam-Warburton cellular automaton of A147562.
It appears that A038573 gives the number of cells turned ON at n-th stage.
Conjecture: a(n) is also the total number of Y-toothpicks after n-th stage in the tree that arises from one of the three spokes in a 120-degree sector of the cellular automaton of A266532 on the triangular grid.
Note that the structure of A266532 is also the "outward" version of the Y-toothpick cellular automaton of A160120.
It appears that A038573 also gives the number of Y-toothpicks added at n-th stage.
Comment from N. J. A. Sloane, Jan 23 2016: All the above conjectures are true!
From Gus Wiseman, Mar 31 2019: (Start)
a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)  (1,5)
                (2,3)  (2,3)  (2,2)  (2,2)
                (3,3)  (3,3)  (2,3)  (2,3)
                       (4,4)  (3,3)  (2,6)
                              (4,4)  (3,3)
                              (4,5)  (4,4)
                              (5,5)  (4,5)
                                     (4,6)
                                     (5,5)
                                     (6,6)
(End)

David Applegate, The movie version
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. A000120, A038573, A147562, A160120, A160720, A161336, A169779, A266532, A266534, A266536.
Cf. A006218, A019565, A070939, A080572, A267610, A267700.
Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325124.

Accumulate[Table[2^DigitCount[n,2,1]-1,{n,0,30}]] (* based on conjecture confirmed by Sloane, Gus Wiseman, Mar 31 2019 *)

a(n) = (A160720(n+1) - 1)/4.
Conjecture 1: a(n) = (A266532(n+1) - 1)/3.
Conjecture 2: a(n) = A160720(n+1) - A266532(n+1).
All of the above conjectures are true. -  N. J. A. Sloane, Jan 23 2016
(Conjecture) a(n) = A267610(n) + n. - Gus Wiseman, Mar 31 2019

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

Original entry on oeis.org

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

A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 133, 225, 237, 273, 309, 417, 453, 561, 669, 961, 973, 1009, 1045, 1153, 1189, 1297, 1405, 1729, 1765, 1873, 1981, 2305, 2413, 2737, 3061, 3969, 3981, 4017, 4053, 4161, 4197, 4305, 4413, 4737, 4773, 4881, 4989, 5313, 5421, 5745

Offset: 0

Omar E. Pol, May 20 2009

The structure has a fractal behavior similar to the toothpick sequence A139250.
First differences: A161415, where there is an explicit formula for the n-th term.
For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  133,  225;
237,  273,  309,  417,  453, 561,  669,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A256530 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _   _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  | |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_| |_|  _ _  |_| |
.          |   | |  _|_|_ _ _|_|_  | |   |
.          |  _| |_|  _ _   _ _  |_| |_  |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _| | |_|  _  |_| | |_ _| |
.          |  _ _  |  _| |_| |_  |  _ _  |
.          | |  _|_| | |_ _ _| | |_|_  | |
.          | |_|  _| |_ _| |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_   _| |_ _| |_  |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _| |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.
(End)

Paolo Xausa, Table of n, a(n) for n = 0..10000
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
Omar E. Pol, Illustration of the structure after 24th stage (contains 1729 ON cells)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.

read("transforms") ; isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc:
A160414 := proc(n) add( A161415(k),k=1..n) ; end proc: seq(A160414(n),n=0..90) ; # R. J. Mathar, Oct 16 2010

A160414list[nmax_]:=Accumulate[Table[If[n<2,n,4*3^DigitCount[n-1,2,1]-If[IntegerQ[Log2[n]],2n,0]],{n,0,nmax}]];A160414list[100] (* Paolo Xausa, Sep 01 2023, after R. J. Mathar *)

my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<Altug Alkan, Sep 25 2015

a(n) = 1 + 4*A219954(n), n >= 1. - M. F. Hasler, Dec 02 2012

a(2^k) = (2^(k+1) - 1)^2. - Omar E. Pol, Jan 05 2013

Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009

More terms from R. J. Mathar, Oct 16 2010

A160118 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 13, 41, 45, 73, 85, 169, 173, 201, 213, 297, 309, 393, 429, 681, 685, 713, 725, 809, 821, 905, 941, 1193, 1205, 1289, 1325, 1577, 1613, 1865, 1973, 2729, 2733, 2761, 2773, 2857, 2869, 2953, 2989, 3241, 3253, 3337, 3373, 3625, 3661, 3913, 4021, 4777, 4789

Offset: 0

Omar E. Pol, May 05 2009

nonn

On the infinite square grid, we start at stage 0 with all square cells in the OFF state.
Define a "peninsula cell" to a cell that is connected to the structure by exactly one of its vertices.
At stage 1 we turn ON a single cell in the central position.
For n>1, if n is even, at stage n we turn ON all the OFF neighboring cells from cells that were turned in ON at stage n-1.
For n>1, if n is odd, at stage n we turn ON all the peninsular OFF cells.
For the corresponding corner sequence, see A160796.
An animation will show the fractal-like behavior (cf. A139250).
For the first differences see A160415. - Omar E. Pol, Mar 21 2011
First differs from A188343 at a(13). - Omar E. Pol, Mar 28 2011

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.888.888.888.888.
.878.878.878.878.
.8866688.8866688.
...656.....656...
.8866444.4446688.
.878.434.434.878.
.888.4422244.888.
.......212.......
.888.4422244.888.
.878.434.434.878.
.8866444.4446688.
...656.....656...
.8866688.8866688.
.878.878.878.878.
.888.888.888.888.
9...............9
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON eight "2" around the central cell, leading to a(2)=a(1)+8=9. In the third generation, four "3" are turned ON at the vertices of the square, a(3)=a(2)+4=13. And so on...

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 cellular automata.

Cf. A139250, A139251, A147562, A147610, A160117, A160119, A160379, A160415, A160796, A188343.

With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

From Nathaniel Johnston, Mar 24 2011: (Start)
a(2n-1) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n-1} A147610(k), n >= 2.
a(2n) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n} A147610(k), n >= 1.
(End)

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010, Feb 21 2010
a(8) - a(38) from Nathaniel Johnston, Nov 06 2010
a(13) corrected at the suggestion of Sean A. Irvine. Then I corrected 19 terms between a(14) and a(38). Finally I added a(39)-a(42). - Omar E. Pol, Mar 21 2011
Rule, for n even, edited by Omar E. Pol, Mar 22 2011
Incorrect comment (in "formula" section) removed by Omar E. Pol, Mar 23 2011, with agreement of author.
More terms from Amiram Eldar, Aug 01 2023

A160720 Number of "ON" cells at n-th stage in 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 9, 21, 25, 37, 49, 77, 81, 93, 105, 133, 145, 173, 201, 261, 265, 277, 289, 317, 329, 357, 385, 445, 457, 485, 513, 573, 601, 661, 721, 845, 849, 861, 873, 901, 913, 941, 969, 1029, 1041, 1069, 1097, 1157, 1185, 1245, 1305, 1429, 1441, 1469, 1497

Offset: 0

Omar E. Pol, May 25 2009

nonn

We work on the vertices of the square grid Z^2, and define the neighbors of a cell to be the four closest cells along the diagonals.
We start at stage 0 with all cells in OFF state.
At stage 1, we turn ON a single cell at the origin.
Once a cell is ON it stays ON.
At each subsequent stage, a cell in turned ON if exactly one of its neighboring cells that are no further from the origin is ON.
The "no further from the origin" condition matters for the first time at stage 8, when only A160721(8) = 28 cells are turned ON, and a(8) = 77. In contrast, A147562(8) = 85, A147582(8) = 36.
This CA also arises as the cross-section in the (X,Y)-plane of the CA in A151776.
In other words, a cell is turned ON if exactly one of its vertices touches an exposed vertex of a ON cell of the previous generation. A special rule for this sequence is that every ON cell has only one vertex that should be considered not exposed: its nearest vertex to the center of the structure.
Analog to the "outward" version (A266532) of the Y-toothpick cellular automaton of A160120 on the triangular grid, but here we have ON cells on the square grid. See also the formula section. - Omar E. Pol, Jan 19 2016
This cellular automaton can be interpreted as the outward version of the Ulam-Warburton two-dimensional cellular automaton (see A147562). - Omar E. Pol, Jun 22 2017

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.8.8.8.8.8.8.8.8.
..7...7...7...7..
.8.6.6.....6.6.8.
....5.......5....
.8.6.4.4.4.4.6.8.
..7...3...3...7..
.8...4.2.2.4...8.
........1........
.8...4.2.2.4...8.
..7...3...3...7..
.8.6.4.4.4.4.6.8.
....5.......5....
.8.6.6.....6.6.8.
..7...7...7...7..
.8.8.8.8.8.8.8.8.
9...............9

JungHwan Min, Table of n, a(n) for n = 0..5000
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.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 30.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A147562, A159912, A160117, A160118, A160120, A160721, A160740, A147562, A160410, A160414, A160412, A160416, A160796, A266532, A267700.

cellOn := [[0,0]] : bbox := [0,0,0,0]: # llx, lly, urx, ury isOn := proc(x,y,L) local i ; for i in L do if op(1,i) = x and op(2,i) = y then RETURN(true) ; fi; od: RETURN(false) ; end: bb := proc(L) local mamin,i; mamin := [0,0,0,0] ; for i in L do mamin := subsop(1=min(op(1,mamin),op(1,i)),mamin) ; mamin := subsop(2=min(op(2,mamin),op(2,i)),mamin) ; mamin := subsop(3=max(op(1,mamin),op(1,i)),mamin) ; mamin := subsop(4=max(op(2,mamin),op(2,i)),mamin) ; od: mamin ; end: for gen from 2 to 80 do nGen := [] ; print(nops(cellOn)) ; for x from op(1,bbox)-1 to op(3,bbox)+1 do for y from op(2,bbox)-1 to op(4,bbox)+1 do # not yet in list? if not isOn(x,y,cellOn) then
# loop over 4 neighbors of (x,y) non := 0 ; for dx from -1 to 1 by 2 do for dy from -1 to 1 by 2 do # test of a neighbor nearer to origin if x^2+y^2 >= (x+dx)^2+(y+dy)^2 then if isOn(x+dx,y+dy,cellOn) then non := non+1 ; fi; fi; od: od: # exactly one neighbor on: add to nGen if non = 1 then nGen := [op(nGen), [x,y]] ; fi; fi; od: od: # merge old and new generation cellOn := [op(cellOn),op(nGen)] ; bbox := bb(cellOn) ; od: # R. J. Mathar, Jul 14 2009

A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (Function[pos, CellularAutomaton[{FromDigits[Boole[#[[2, 2]] == 1 || Count[Flatten[#], 1] == 1 && Count[Extract[#, pos], 1] == 1] & /@ Tuples[{1, 0}, {3, 3}], 2], 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}]] /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{#, 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}] & /@ {13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900810351549134401372178, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940267935557047531012112, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940286382301121240563712, 13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900828798293208110923778})], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{46, {2, ReplacePart[ArrayPad[{{1}}, 1], # -> 2]}, {1, 1}}, {{{1}}, 0}, {{{m}}, All, All}] & /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 24 2016 *)

Conjecture: a(n) = 1 + 4*(A266532(n) - 1)/3, n >= 1. - Omar E. Pol, Jan 19 2016. This formula is correct! - N. J. A. Sloane, Jan 23 2016

a(n) = 1 + 4*A267700(n-1) = 1 + 2*(A159912(n) - n), n >= 1. - Omar E. Pol, Jan 24 2016

Edited by N. J. A. Sloane, Jun 26 2009

More terms from David Applegate, Jul 03 2009

A079314 Number of first-quadrant cells (including the two boundaries) born at stage n of the Holladay-Ulam cellular automaton.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 10, 2, 4, 4, 10, 4, 10, 10, 28, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 4, 10, 10, 28, 10, 28, 28, 82, 10, 28, 28, 82, 28, 82, 82, 244, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 4

Offset: 0

N. J. A. Sloane, Feb 12 2003

See the main entry for this CA, A147562, for further information.
When I first read the Singmaster MS in 2003 I misunderstood the definition of the CA. In fact once cells are ON they stay ON. The other version, when cells can change state from ON to OFF, is described in A079317. - N. J. A. Sloane, Aug 05 2009
The pattern has 4-fold symmetry; sequence just counts cells in one quadrant.

			From _Omar E. Pol_, Jul 18 2009: (Start)
If written as a triangle:
  1;
  2;
  2,4;
  2,4,4,10;
  2,4,4,10,4,10,10,28;
  2,4,4,10,4,10,10,28,4,10,10,28,10,28,28,82;
  2,4,4,10,4,10,10,28,4,10,10,28,10,28,28,82,4,10,10,28,10,28,28,82,10,28;...
Rows converge to A151712.
(End)

D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.

Paolo Xausa, Table of n, a(n) for n = 0..10000
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 (Overlapping squares) [From _Omar E. Pol_, Nov 20 2009]
D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Cf. A147582, A079315, A079316, A079317, A079318, A079319, A151713, A139250.

Cf. A151712, A151922, A160407.

A079314list[nmax_]:=Join[{1},3^(DigitCount[Range[nmax],2,1]-1)+1];A079314list[100] (* Paolo Xausa, Jun 29 2023 *)

For n > 0, a(n) = 3^(A000120(n)-1) + 1.
For n > 0, a(n) = A147582(n)/4 + 1.
Partial sums give A151922. [Omar E. Pol, Nov 20 2009]

Edited by N. J. A. Sloane, Aug 05 2009

cp's OEIS Frontend

A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A187220 Gullwing sequence (see Comments lines for precise definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple