A170905 -id:A170905 - cp's OEIS frontend

A169778 a(n) = ceiling(A170905/2).

0, 1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 13, 9, 4, 7, 12, 14, 10, 16, 16, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 4, 7, 12, 14, 14, 22, 34, 30, 14, 16, 28, 35, 25, 35, 32, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21

Offset: 0

N. J. A. Sloane, May 08 2010

The hexagon-based CA of A151723 has as symmetry group the dihedral group of order 12. Consider a one-twelfth slice; a(n) is the number of cells that are turned from OFF to ON at generation n.

			Illustration of generations 1 through 9:
.1
..2
...3
..4.4
.....5
..7.6.6
...7...7
..8.8.8.8
.........9
...
From _Omar E. Pol_, Feb 12 2013: (Start)
When written as a triangle from 1, the right border gives A011782 and row lengths give A011782.
1,
1,
1,2,
1,2,3,4,
1,2,3,5,5,4,7,8,
1,2,3,5,5,5,9,13,9,4,7,12,14,10,16,16,
1,2,3,5,5,5,9,13,9,5,9,15,19,17,21,29,17,4,7,12,14,14,22,34,30,14,16,28,35,25,35,32;
1,2,3,5,5,5,9,13,9,5,9,15,19,17,21,... (End)

Cf. A151723, A151724, A170905.

A170905(n) = 2a(n) except A170905(1)=1.

A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 55, 85, 127, 133, 151, 181, 235, 289, 331, 409, 499, 505, 523, 553, 607, 661, 715, 817, 967, 1069, 1111, 1189, 1327, 1489, 1603, 1789, 1975, 1981, 1999, 2029, 2083, 2137, 2191, 2293, 2443, 2545, 2599, 2701, 2875, 3097, 3295

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

Analog of A151725, but here we are working on the triangular lattice (or the A_2 lattice) where each hexagonal cell has six neighbors.
A cell is turned ON if exactly one of its six neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
It would be nice to find a recurrence for this sequence!
Has a behavior similar to A182840 and possibly to A182632. - Omar E. Pol, Jan 15 2016

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

N. J. A. Sloane, Table of n, a(n) for n = 0..4095 [First 1026 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
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.]
Bradley Klee, Log-periodic coloring, over the half-hexagon tiling.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.

A151723[0] = 0; A151723[n_] := Total[CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151723, 47, 0](* JungHwan Min, Sep 01 2016 *)
A151723L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151723L[46] (* JungHwan Min, Sep 01 2016 *)

a(n) = 6*A169780(n) - 6*n + 1 (this is simply the definition of A169780).
a(n) = 1 + 6*A169779(n-2), n >= 2. - Omar E. Pol, Mar 19 2015
It appears that a(n) = a(n-2) + 3*(A256537(n) - 1), n >= 3. - Omar E. Pol, Apr 04 2015

Edited by N. J. A. Sloane, Jan 10 2010

A169780 Total number of ON cells after n-th stage in one-sixth slice of hexagonal CA defined in A151723 (including both boundaries).

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, 185, 203, 211, 225, 249, 277, 297, 329, 361, 363, 367, 373, 383, 393, 403, 421, 447, 465, 475, 493, 523, 561, 595, 637, 695, 729, 737, 751, 775, 803, 831, 875, 943, 1003, 1031

Offset: 0

N. J. A. Sloane, May 11 2010

nonn

Partial sums of A170905.

N. J. A. Sloane, Table of n, a(n) for n = 0..1025
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A151723, A169779, A169781.

a(n) = n + (A151723(n) - 1)/6, n >= 1. - Omar E. Pol, Mar 06 2013

A170898 Triangle read by rows, obtained by dividing A151724 by 6.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 9, 9, 7, 13, 15, 1, 3, 5, 9, 9, 9, 17, 25, 17, 7, 13, 23, 27, 19, 31, 31, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 7, 13, 23, 27, 27, 43, 67, 59, 27, 31, 55, 69, 49, 69, 63, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41

Offset: 0

N. J. A. Sloane, Jan 10 2010

Row k has 2^k terms.

Right border gives the positive terms of A000225. - Omar E. Pol, Sep 28 2013

			Triangle begins:
1;
1,3;
1,3,5,7;
1,3,5,9,9,7,13,15;
1,3,5,9,9,9,17,25,17,7,13,23,27,19,31,31;
1,3,5,9,9,9,17,25,17,9,17,29,37,33,41,57,33,7,13,23,27,27,43,67,59,27,31,55,69,49,69,63;
...

N. J. A. Sloane, Table of n, a(n) for n = 0..1023

Cf. A169779 (partial sums).

Cf. A139250, A139251, A151723, A151724, A170899.

Equals A170905(n) - 1.

A233781 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the D-toothpick "wide" triangle of A233780.

Original entry on oeis.org

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, 18, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 18, 10, 18, 30, 38, 26, 42, 52, 34, 8, 14, 22, 26, 22, 42, 56, 50, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10

Offset: 0

Omar E. Pol, Dec 15 2013

Essentially the first differences of A233780.
First differs from A170905 at a(24).
First differs from A233971 at a(25).
First differs from A233765 at a(44).

			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;
2,4,6,10,10,10,18,24,18,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,18,10,18,30,38,26,42,52,34,8,14,22,26,22,42,56,50,16,30,46,58,32,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A170905, A220501, A220521, A231349, A233765, A233780, A233782, A233971.

A170906 Triangle read by rows: T(n,k) = number of cells that are turned from OFF to ON at stage k of the cellular automaton in the 30-60-90 triangle of hexagons defined in Comments.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 4, 1, 2, 1, 1, 2, 2, 4, 2, 2, 3, 3, 1, 1, 2, 2, 4, 2, 4, 5, 4, 1, 2, 1, 1, 2, 2, 4, 2, 4, 6, 6, 1, 2, 3, 3, 1, 1, 2, 2, 4, 2, 4, 6, 8, 1, 2, 3, 5, 3, 3, 1, 1, 2, 2, 4, 2, 4, 6, 8, 2, 2, 3, 5, 5, 3, 5, 4, 1, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 5, 6, 7, 6, 6, 4, 1, 2, 1

Offset: 1

N. J. A. Sloane, Jan 24 2010

Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905.
Assume the hexagons are oriented so that each one has a pair of vertical edges.
Consider the (30 deg., 60 deg., 90 deg.) triangle of hexagons with n hexagons along the short side, along the X-axis, 2n-1 hexagons along the hypotenuse and n hexagons separated by single edges along the middle side, along the Y-axis.
Initially all cells are OFF. At stage 1, the cell in the 60-degree corner is turned ON; thereafter, a cell is turned ON if it has exactly one ON neighbor in the triangle. Once a cell is ON it stays ON.
T(n,k) is the number of cells that are turned from OFF to ON at stage k (1 <= k <= 2n-1).
The rows converge to A170905. The rows sums give A170907.
Row n contains 2n-1 terms.
I wish I had a recurrence for this sequence!

			Triangle begins:
1
1 2 1
1 2 2 2 1
1 2 2 4 1 2 1
1 2 2 4 2 2 3 3 1
1 2 2 4 2 4 5 4 1 2 1
1 2 2 4 2 4 6 6 1 2 3 3 1
1 2 2 4 2 4 6 8 1 2 3 5 3 3 1
1 2 2 4 2 4 6 8 2 2 3 5 5 3 5 4 1
1 2 2 4 2 4 6 8 2 4 5 6 7 6 6 4 1 2 1
...
Row n = 4, [1 2 2 4 1 2 1], corresponds to the sequence of cells being turned ON shown in the following triangle (X denotes a cell that stays OFF). The hexagons have to be imagined.
7
.6
6.5
.X.4
X.4.3
.4.X.2
4.3.2.1

N. J. A. Sloane, Table of n, a(n) for n = 1..16384 (Rows 1..128, flattened)
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. A151723, A151724, A170905, A170907, A169782 (partial sums across rows).

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

Original entry on oeis.org

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.

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.

A169788 a(n) = n(n+1)/2 - A169780(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 6, 7, 7, 14, 20, 25, 27, 30, 36, 37, 37, 52, 66, 79, 89, 100, 112, 117, 115, 122, 140, 153, 157, 158, 168, 167, 167, 198, 228, 257, 283, 310, 338, 359, 373, 396, 428, 453, 467, 474, 486, 491, 481, 496, 538, 575, 603, 628, 654, 665, 653, 650, 680, 707, 711

Offset: 0

N. J. A. Sloane, May 13 2010

nonn

If n is a power of 2, this is the number of OFF cells after n stages in a 60-degree wedge of the hexagonal CA (see A170905, A169780, A151723, A169789).

N. J. A. Sloane, Table of n, a(n) for n = 0..1025

Cf. A170905, A169780, A151723, A169789.

A233765 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the toothpick "wide" triangle of A233764.

Original entry on oeis.org

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, 18, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 18, 10, 18, 28, 38, 30, 50, 54, 34, 8, 14, 22, 26, 22, 42, 56, 50, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10

Offset: 0

Omar E. Pol, Dec 16 2013

Essentially the first differences of A233764.
First differs from A170905 at a(24).
First differs from A233971 at a(25).
First differs from A233781 at a(44).

			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;
2,4,6,10,10,10,18,24,18,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,18,10,18,28,38,30,50,54,34,8,14,22,26,22,42,56,50,16,30,46,58,32,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A220501, A220521, A231349, A233764, A233781, A233971.

cp's OEIS Frontend

A169778 a(n) = ceiling(A170905/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A169780 Total number of ON cells after n-th stage in one-sixth slice of hexagonal CA defined in A151723 (including both boundaries).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A170898 Triangle read by rows, obtained by dividing A151724 by 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A233781 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the D-toothpick "wide" triangle of A233780.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A170906 Triangle read by rows: T(n,k) = number of cells that are turned from OFF to ON at stage k of the cellular automaton in the 30-60-90 triangle of hexagons defined in Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A169788 a(n) = n(n+1)/2 - A169780(n).

Views

Author

Keywords

Comments

Links

Crossrefs

A233765 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the toothpick "wide" triangle of A233764.