A233970 -id:A233970 - cp's OEIS frontend

A233971 Number of toothpicks added at n-th stage to the structure of A233970.

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 24, 22, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 22, 10, 18, 28, 38, 28, 46, 56, 54, 8, 14, 22, 26, 22, 42, 56, 62, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10

Offset: 0

Omar E. Pol, Dec 18 2013

Essentially the first differences of A233970.
First differs from A170905 at a(24).
First differs from both A233765 and A233781 at a(25).

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,24,22,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,22,10,18,28,38,28,46,56,54,8,14,22,26,22,42,56,62,16,30,46,58,32,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A161207, A161831, A170905, A182633, A182635, A182841, A233761, A233765, A233781, A233970, A233972.

A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 61, 77, 91, 105, 127, 143, 147, 153, 165, 181, 197, 217, 249, 285, 307, 321, 349, 391, 431, 467, 517, 549, 553, 559, 571, 587, 603, 623, 655, 691, 715

Offset: 0

Omar E. Pol, Dec 12 2010

An H-toothpick sequence is a toothpick sequence on a square grid that resembles a partial honeycomb of hexagons.
The structure has two types of elements: the classic toothpicks with length 1 and the "D-toothpicks"  with length sqrt(2).
Classic toothpicks are placed in the vertical direction and D-toothpicks are placed in a diagonal direction.
Each hexagon has area = 4.
The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.
The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first differences) gives the number added at the n-th stage.
[It appears that for this sequence a classic toothpick is a line segment of length 1 that is parallel to the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. It also appears that classic toothpicks are not placed on the y-axis. - N. J. A. Sloane, Feb 06 2023]
From Omar E. Pol, Feb 17 2023: (Start)
This cellular automaton appears to be a version on the square grid of the first quadrant of the structure of A182840.
The rules are as follows:
- The elements (toothpicks and D-toothpicks) are connected at their ends.
- At each free end of the elements of the old generation two elements of the new generation must be connected.
- The toothpicks of length 1 must always be placed vertically, i.e. parallel to the Y-axis.
- The angle between a toothpick of length 1 and a D-toothpick of length sqrt(2) that share the same node must be 135 degrees, therefore the angle between two D-toothpicks that share the same node is 90 degrees.
As a result of these rules we can see that in the odd-indexed rows of the structure are placed only the toothpicks of length 1 and in the even-indexed rows of the structure are placed the D-toothpicks of length sqrt(2).
Apart from the trapezoids, pentagons and heptagons that are adjacent to the axes of the first quadrant it appears that there are only three types of polygons:
- Regular hexagons of area 4.
- Concave decagons (or concave 10-gons) of area 8.
- Concave dodecagons (or concave 12-gons) of area 12.
There are infinitely many of these polygons.
The structure shows a fractal-like behavior as we can see in other members of the family of toothpick cellular automata.
The structure has internal growth as some members of the mentioned family. (End)

			We start at stage 0 with no toothpicks.
At stage 1 we place a D-toothpick [(0,1),(1,2)], so a(1)=1.
At stage 2 we place a toothpick [(1,2),(1,3)] and a D-toothpick [(1,2),(2,1)], so a(2)=1+2=3.
At stage 3 we place 4 elements: a D-toothpick [(1,3),(0,4)], a D-toothpick [(1,3),(2,4)], a D-toothpick [(2,1),(3,2)] and a toothpick [(2,1),(2,0)], so a(3)=3+4=7. Etc.
The first hexagon appears in the structure after 4 stages.

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 a(32) = 549
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A153000, A161206, A170888, A172308, A182632, A182634, A182839, A182840, A187212, A194444, A220524, A233970, A267458, A267694, A267698.

See A360501 and A360512 for another hexagonal net built on the square grid. - N. J. A. Sloane, Feb 09 2023

Conjecture: a(n) = (A182840(n+1) + A267458(n+1) - 2)/4. - Omar E. Pol, Feb 10 2023

Partially edited by N. J. A. Sloane, Feb 06 2023

a(19)-a(41) from Omar E. Pol, Feb 06 2023

A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 509, 547, 573, 615, 667, 701, 709, 723, 745, 771

Offset: 0

Omar E. Pol, Dec 15 2013

nonn

The D-toothpicks placed in northwest or northeast direction both are prohibited, except in the substructures in which the symmetry could be broken, so a(44) = 509, not 507. For another version with broken symmetry in some substructures see A233764. See also A231348, a simpler cellular automaton based in triangles which has essentially a similar structure.
A233781 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233764 at a(44).

Cf. A139250, A169780, A220500, A231348, A233764, A233781, A233970.

A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).

Original entry on oeis.org

1, 3, 4, 8, 11, 13, 14, 22, 29, 35, 40, 44, 47, 49, 50, 66, 81, 95, 108, 120, 131, 141, 150, 158, 165, 171, 176, 180, 183, 185, 186, 218, 249, 279, 308, 336, 363, 389, 414, 438, 461, 483, 504, 524, 543, 561, 578, 594, 609, 623, 636, 648, 659, 669, 678, 686, 693, 699, 704, 708, 711, 713, 714, 778, 841, 903, 964, 1024

Offset: 1

Omar E. Pol, Mar 30 2015

Also partial sums of A080079.
In order to construct the structure we use the following rules:
On the infinite triangular grid we are in a 60-degree wedge with the vertex located on top of the wedge.
The nearest triangular cell to the vertex remains OFF.
At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.
At stage n, in the n-th level of the structure, we turn ON k cells connected by their vertices with their bases up, where k = A080079(n).
The cells turned ON remain ON forever.
The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
Note that this is also the structure in every one of the six wedges of the structure of A256266.
A080079 gives the number of cells turned ON at n-th stage.

			Illustration of initial terms:
-----------------------------------------------------------
n   A080079(n)   a(n)                  Diagram
-----------------------------------------------------------
.                                        / \
1       1         1                     / T \
2       2         3                    / T T \
3       1         4                   /   T   \
4       4         8                  / T T T T \
5       3        11                 /   T T T   \
6       2        13                /     T T     \
7       1        14               /       T       \
8       8        22              / T T T T T T T T \
9       7        29             /   T T T T T T T   \
10      6        35            /     T T T T T T     \
11      5        40           /       T T T T T       \
12      4        44          /         T T T T         \
13      3        47         /           T T T           \
14      2        49        /             T T             \
15      1        50       /               T               \
...
For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
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. 37.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A047999, A001316, A080079, A139250, A169779, A169788, A170905, A233970, A256256, A256266.

Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 6}] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = A256266(n)/6.

A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

Offset: 0

Omar E. Pol, Dec 16 2013

nonn

The D-toothpicks placed in northwest or northeast direction both are prohibited. Note that due this rule there are substructures with broken symmetry, for instance a(44) = 507, not 509. For another version without broken symmetry see A233780.
A233765 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233780 at a(44).

Cf. A139250, A169780, A220500, A220520, A231348, A233765, A233780, A233970.

A233972 a(n) = A233971(n+1)/2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 12, 11, 4, 7, 11, 13, 8, 15, 16, 1, 2, 3, 5, 5, 5, 9, 12, 11, 5, 9, 14, 19, 14, 23, 28, 27, 4, 7, 11, 13, 11, 21, 28, 31, 8, 15, 23, 29, 16, 31, 32, 1, 2, 3, 5, 5, 5, 9, 12, 11, 5, 9, 14

Offset: 1

Omar E. Pol, Dec 18 2013

			Written as an irregular triangle in which row lengths is A000079 the sequence begins:
1,
1,2,
1,2,3,4,
1,2,3,5,5,4,7,8,
1,2,3,5,5,5,9,12,11,4,7,11,13,8,15,16,
1,2,3,5,5,5,9,12,11,5,9,14,19,14,23,28,27,4,7,11,13,11,21, 28,31,8,15,23,29,16,31,32;

Right border gives A000079.

Cf. A139251, A233782, A233970, A233971.

cp's OEIS Frontend

A233971 Number of toothpicks added at n-th stage to the structure of A233970.

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A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).

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A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

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A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).

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A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

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A233972 a(n) = A233971(n+1)/2.

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