A151724 -id:A151724 - cp's OEIS frontend

A170905 Consider the hexagonal cellular automaton defined in A151723, A151724; a(n) = number of cells that go from OFF to ON at stage n, if we only look at a 60-degree wedge (including the two bounding edges).

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 26, 18, 8, 14, 24, 28, 20, 32, 32, 2, 4, 6, 10, 10, 10, 18, 26, 18, 10, 18, 30, 38, 34, 42, 58, 34, 8, 14, 24, 28, 28, 44, 68, 60, 28, 32, 56, 70, 50, 70, 64, 2, 4, 6, 10, 10, 10, 18, 26, 18, 10, 18, 30, 38, 34, 42

Offset: 0

N. J. A. Sloane, Jan 22 2010

			From _Omar E. Pol_, Feb 12 2013: (Start)
When written as a triangle starting from 1, the right border gives A000079 and row lengths give A011782.
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,26,18,8,14,24,28,20,32,32;
2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,58,34,8,14,24,28,28,44,68,60,28,32,56,70,50,70,64;
2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,...
... (End)

N. J. A. Sloane, Table of n, a(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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A151723, A151724, A170898, A169778, A169780 (partial sums).

a(n) = A170898(n-2) + 1 for n >= 2.

a(n) = 2*A169778(n) for n != 1.

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.

A169759 Sum of terms in n-th row of A151724 when that sequence is written as a triangle.

Original entry on oeis.org

0, 1, 6, 24, 96, 372, 1476, 5880, 23472, 93792, 374964, 1499424

Offset: 0

N. J. A. Sloane, May 06 2010

nonn

I wish I had a recurrence for this sequence!

Cf. A151724, A169760.

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

A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 4, 3, 6, 7, 0, 1, 2, 4, 4, 4, 8, 12, 8, 3, 6, 11, 13, 9, 15, 15, 0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 3, 6, 11, 13, 13, 21, 33, 29, 13, 15, 27, 34, 24, 34, 31, 0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 4, 8, 14

Offset: 0

N. J. A. Sloane, Jan 10 2010

This sequence is essentially the number of cells that are turned ON at the n-th generation of a 30-degree sector of the hexagonal Ulam-Warburton cellular automaton in A151723. The cells on the six main diagonals are ignored, and the resulting counts have been divided by 12. - N. J. A. Sloane, Mar 13 2021
Row k has 2^k terms.
It would be nice to have a formula or recurrence for any of A170899, A342272-A342278, or any nontrivial relation between them. This might help to understand the fractal structure of the mysterious hexagonal Ulam-Warburton cellular automaton A151723. - N. J. A. Sloane, Mar 14 2021
It appears that this may also be regarded as a tetrahedron E(m,i,j), m>=0, i>=0, j>=0, in which the slice m is a triangle read by rows: R(i,j) in which row i has length A011782(i). - Omar E. Pol, Feb 13 2013
It appears that in the slice m (of the tetrahedron mentioned above) the differences between the first 2^(m-3) elements of row m-1 and the first 2^(m-3) elements of row m give the first 2^(m-3) elements of A169787, if m >= 3. Also it appears that the right border of slice m gives the first m powers of 2 together with 0. See the second arrangement in Example section. - Omar E. Pol, Mar 16 2013

			Triangle begins:
0;
0,1;
0,1,2,3;
0,1,2,4,4,3,6,7;
0,1,2,4,4,4,8,12,8,3,6,11,13,9,15,15;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,3,6,11,13,13,21, 33,29,13,15,27,34,24,34,31;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32,3,6,11,13,13,21,33,29,17,21, 37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58, 73,63;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32,4,8,14,18,18,26,42,42,26,26, 46,66,70,74,98,90,40,20,36,50,54,70,110,126,86,58,86,124, 118,118,132,64,3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127;
...
From _Omar E. Pol_, Feb 13 2013 (Start):
When written as a tetrahedron the slices 0-7 are:
0;
..
1;
0;
..
1;
2;
3,0;
....
1;
2;
4,4;
3,6,7,0;
........
1;
2;
4,4;
4,8,12,8;
3,6,11,13,9,15,15,0;
....................
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
3,6,11,13,13,21,33,29,13,15,27,34,24,34,31,0;
.............................................
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58,73,63,0;
..........................................................
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;
4,8,14,18,18,26,42,42,26,26,46,66,70,74,98,90,40,20,36,50,54,70,110,126,86,58,86,124,118,118,132,64;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127,0;
..........................................................
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..4092

Cf. A139250, A151723, A151724, A170898.

A342272, A342273, A342274 are limiting sequences to which various parts of the rows of this triangle converge.

A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

Original entry on oeis.org

0, 1, 4, 8, 14, 16, 14, 24, 38, 32, 14, 24, 46, 64, 54, 56, 86, 64, 14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128, 14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256

Offset: 0

Omar E. Pol, Dec 09 2010

First differences of A182840.

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle:
0;
1;
4;
8;
14, 16;
14, 24, 38, 32;
14, 24, 46, 64, 54, 56, 86, 64;
14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128;
14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256;
...
(End)

Olaf Voß, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Olaf Voß, Toothpick structures on hexagonal net
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160121, A161207, A182633, A182635, A151724, A182842.

a(2^k + 1) = 2^(k+2), at least for 0 <= k <= 9. - Omar E. Pol, Nov 01 2014

More terms from Olaf Voß, Dec 24 2010

Wiki link added by Olaf Voß, Jan 14 2011

A342272 The rows of the triangle A170899 converge to this sequence.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 4, 8, 14, 18, 18, 26, 42, 42, 24, 20, 36, 50, 46, 50, 62, 32, 4, 8, 14, 18, 18, 26, 42, 42, 26, 26, 46, 66, 70, 74, 98, 90, 40, 20, 36, 50, 54, 70, 110, 126, 86, 58, 86, 124, 118, 118, 132

Offset: 0

N. J. A. Sloane, Mar 13 2021

nonn

It would be nice to have a formula or recurrence for any of A170899, A342272-A342278, or any nontrivial relation between them. This might help to understand the fractal structure of the mysterious hexagonal Ulam-Warburton cellular automaton A151723.
A342273 is the limiting sequence for the part of the row of A170899 that start at the first "3".
Needs a bigger b-file.

N. J. A. Sloane, Table of n, a(n) for n = 0..512

Cf. A151723, A151724, A170899, A342273, A342274.

This is A169787 with 1 subtracted from each term.

A342273 Consider the k-th row of triangle A170899 starting at the 3 in the middle of the row; the row from that point on converges to this sequence as k increases.

Original entry on oeis.org

3, 6, 11, 13, 13, 21, 33, 29, 17, 21, 37, 51, 51, 57, 77, 61, 25, 21, 37, 51, 55, 71, 111, 127, 91, 65, 93, 137, 143, 147, 175, 127, 41, 21, 37, 51, 55, 71, 111, 127, 95, 79, 119, 179, 207, 219, 271, 279, 171, 81, 93, 137, 159, 195, 291, 363

Offset: 0

N. J. A. Sloane, Mar 13 2021

nonn

It would be nice to have a formula or recurrence for any of A170899, A342272-A342278, or any nontrivial relation between them. This might help to understand the fractal structure of the mysterious hexagonal Ulam-Warburton cellular automaton A151723.

Needs a bigger b-file.

			Row k=6 of A170899 breaks up naturally into 7 pieces:
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58,73,63,0.
The last piece already matches the sequence for 16 terms. The number of matching terms doubles at each row.

N. J. A. Sloane, Table of n, a(n) for n = 0..255

Cf. A151723, A151724, A342272.

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).

A222181 Number of pentagons added at n-th stage to the structure of A222180.

Original entry on oeis.org

0, 1, 5, 10, 10, 10, 20, 30, 20, 10, 20, 40, 40, 30, 50, 70, 40, 10, 20, 40, 60, 80

Offset: 0

Omar E. Pol, Mar 15 2013

Essentially the first differences of A222180.

Also number of P-toothpicks added at n-th stage to the P-toothpick structure of A222180.

			Apparently this is an irregular triangle:
0;
1;
5;
10,10;
10,20,30,20;
10,20,40,40,30,50,70,40;
10,20,40,60,80,...

Cf. A139250, A139251, A147582, A151724, A152968, A161645, A182633, A182841, A222173, A222177, A222180.

a(n) = 10*A222173(n-2), n >= 3. - Omar E. Pol, Nov 24 2013

cp's OEIS Frontend

A170905 Consider the hexagonal cellular automaton defined in A151723, A151724; a(n) = number of cells that go from OFF to ON at stage n, if we only look at a 60-degree wedge (including the two bounding edges).

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

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A169759 Sum of terms in n-th row of A151724 when that sequence is written as a triangle.

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

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A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.

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A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

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A342272 The rows of the triangle A170899 converge to this sequence.

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A342273 Consider the k-th row of triangle A170899 starting at the 3 in the middle of the row; the row from that point on converges to this sequence as k increases.

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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.

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A222181 Number of pentagons added at n-th stage to the structure of A222180.

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