A152980 -id:A152980 - cp's OEIS frontend

A151694 G.f.: Product_{k>=1} (1 + 2x^(2^k-1) + 2x^(2^k)).

1, 2, 2, 2, 6, 8, 4, 2, 6, 8, 8, 16, 28, 24, 8, 2, 6, 8, 8, 16, 28, 24, 12, 16, 28, 32, 48, 88, 104, 64, 16, 2, 6, 8, 8, 16, 28, 24, 12, 16, 28, 32, 48, 88, 104, 64, 20, 16, 28, 32, 48, 88, 104, 72, 56, 88, 120, 160, 272, 384, 336, 160, 32, 2, 6, 8, 8, 16, 28, 24, 12, 16, 28, 32, 48, 88

Offset: 0

N. J. A. Sloane, Jun 04 2009

nonn

			From _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
  1;
  2,2;
  2,6,8,4;
  2,6,8,8,16,28,24,8;
  2,6,8,8,16,28,24,12,16,28,32,48,88,104,64,16;
  2,6,8,8,16,28,24,12,16,28,32,48,88,104,64,20,16,28,32,48,88,104,72,56,88,...
(End)

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

For generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.

Cf. A000079. - Omar E. Pol, Jun 09 2009

CoefficientList[Series[Product[1+2x^(2^k-1)+2x^2^k,{k,10}],{x,0,80}],x] (* Harvey P. Dale, Oct 07 2020 *)

A169708 First differences of A169707.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 20, 44, 68, 60, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 20, 44, 68, 76, 84, 156, 196, 140, 84, 156, 212, 236, 324, 508, 516, 252, 4, 12, 20, 28, 20

Offset: 0

N. J. A. Sloane, Apr 17 2010

			From _Omar E. Pol_, Feb 13 2015: (Start)
Written as an irregular triangle in which row lengths are 1,1,2,4,8,16,32,... the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124,20,44,68,76,84,156,196,140,84,156,212,236,324,508,516,252;
It appears that the row sums give A000302.
It appears that the right border gives A173033.
(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

Cf. A000302, A011782, A139251, A151548, A152980, A153006, A160164, A160552, A169707 (partial sums), A170903, A173033, A253088.

It appears that a(n) = 4*A160552(n), n >= 1. - Omar E. Pol, Feb 13 2015

Initial 1 added by Omar E. Pol, Feb 13 2015

A160412 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, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647

Offset: 0

Omar E. Pol, May 20 2009, Jun 01 2009

nonn

From Omar E. Pol, Nov 10 2009: (Start)
On the infinite square grid, consider the outside corner of an infinite square.
We start at round 0 with all cells in the OFF state.
The rule: 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.
At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.
At round 2, we turn ON nine cells around the hexagon.
At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
And so on.
Shows a fractal-like behavior similar to the toothpick sequence A153006.
For the first differences see the entry A162349.
For more information see A160410, which is the main entry for this sequence.
(End)

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......

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 [From _Omar E. Pol_, Nov 10 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata - _Omar E. Pol_, Nov 10 2009

Cf. A139250, A139251, A153006, A152980, A160410, A160414.

Cf. A130665, A162349. - Omar E. Pol, Nov 10 2009

a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)

From Omar E. Pol, Nov 10 2009: (Start)
a(n) = A160410(n)*3/4.
a(0) = 0, a(n) = A130665(n-1)*3, for n>0.
(End)

More terms from Omar E. Pol, Nov 10 2009
Edited by Omar E. Pol, Nov 11 2009
More terms from Nathaniel Johnston, Nov 06 2010
More terms from Colin Barker, Apr 19 2015

A153007 Triangular number A000217(n) minus toothpick number A153006(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 0, 3, 8, 11, 13, 15, 13, 5, 0, 7, 20, 31, 41, 51, 57, 57, 59, 69, 79, 82, 81, 74, 51, 17, 0, 15, 44, 71, 97, 123, 145, 161, 179, 205, 231, 250, 265, 274, 267, 249, 247, 273, 307, 334, 357, 374, 375, 364, 363, 376, 380, 364, 332, 270, 163, 49, 0, 31, 92, 151

Offset: 0

Omar E. Pol, Dec 19 2008, May 27 2009

nonn

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
OEIS (Plot 2), Triangular numbers (A000217) and toothpick numbers (A153006) vs n [From _Omar E. Pol_, Apr 29 2009]

Cf. A000217, A000225, A000396, A000668, A006516, A139250, A139251, A152980, A153006.

a(n) = A000217(n)-A153006(n).

More terms from R. J. Mathar, Jul 13 2009

A159795 a(n) = 4*A153006(n).

Original entry on oeis.org

0, 4, 12, 24, 36, 52, 80, 112, 132, 148, 176, 212, 252, 312, 400, 480, 516, 532, 560, 596, 636, 696, 784, 868, 924, 984, 1076, 1188, 1328, 1536, 1792, 1984, 2052, 2068, 2096, 2132, 2172, 2232, 2320, 2404, 2460, 2520, 2612, 2724, 2864, 3072, 3328, 3524, 3612

Offset: 0

Omar E. Pol, May 02 2009

nonn

For the first differences see A147646. - Omar E. Pol, Jul 24 2009

It appears that a(n) is also the total path length of a toothpick structure as A139250 after n-th stage which is constructed following a special rule: toothpicks of the new generation have length 4 when are placed on the 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 a 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 length 4, 3 and 2. a(n) is also the total number of components in the structure after n-th stage. a(n) is also the number of grid points that are covered after n-th stage, except the central point of the structure. The toothpick sequence A139250 gives the number of toothpicks after n-th stage. - Omar E. Pol, Oct 24 2011

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, A152980, A153006, A159793, A159794.

Cf. A147646, A152980. - Omar E. Pol, Jul 24 2009

More terms from Omar E. Pol, Jul 24 2009

A151555 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 80, 33, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 81, 39, 20, 33, 41, 50, 72, 97, 89, 66, 73, 107, 132, 172, 240, 272, 192, 65, 5, 10, 13, 15, 20, 32, 33, 23

Offset: 0

N. J. A. Sloane, May 20 2009

nonn

From Gary W. Adamson, May 25 2009: (Start)
Convolved with A078008 signed (A151575) [1, 0, 2, -2, 6, -10, 22, -42, 86, -170, ...]
equals the toothpick sequence A153006: (1, 3, 6, 9, 13, 20, 28, ...). (End)
If A151550 is written as a triangle then the rows converge to this sequence. - N. J. A. Sloane, Jun 16 2009

			From _Omar E. Pol_, Jun 19 2009: (Start)
May be written as a triangle:
1;
3;
4,5;
5,10,12,9;
5,10,13,15,20,32,32,17;
5,10,13,15,20,32,33,23,20,33,41,50,72,96,80,33;
5,10,13,15,20,32,33,23,20,33,41,50,72,96,81,39,20,33,41,50,72,97,89,66,73,...
(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

Cf. A139250, A151551, A151552, A151553, A151554, A151550, A152980, A153006.

Cf. A078008. - Gary W. Adamson, May 25 2009

A194693 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194692.

Original entry on oeis.org

2, 2, 4, 6, 8, 8, 12, 13, 12, 10, 16, 20, 32, 20, 24, 27, 20, 10, 16, 24, 36, 41, 48, 48, 54, 34, 44, 50, 80, 48, 48, 55, 36, 10, 16, 24, 36, 41, 48, 54, 68, 62, 80, 80, 124, 95, 104, 89, 94, 44, 48, 68

Offset: 1

Omar E. Pol, Sep 03 2011

Sequence related to the D-toothpick "wide" triangle (See A194440 and A194441). First differences of A194692.

			Written as a triangle:
2,
2,
4,
6,8,
8,12,13,12,
10,16,20,32,20,24,27,20,
10,16,24,36,41,48,48,54,34,44,50,80,48,48,55,36,
10,16,24,36,41,48,54,68,62,80,80,124,95,104,89,94,44,...

Cf. A086224, A152980, A194271, A194440, A194441, A194692, A194695, A194696

a(n) = A194696(n)/2.
Conjectures for n = 2^k+j, if 0<=j<=3:
a(2^k) = 7*2^(k-2)-1, if k >= 2.
a(2^k+1) = 2^k+4, if k >= 2.
a(2^k+2) = 10, if k >= 3.
a(2^k+3) = 16, if k >= 3.
End conjectures.

A194695 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194694.

Original entry on oeis.org

2, 2, 4, 6, 8, 8, 11, 16, 13, 10, 12, 20, 22, 26, 24, 36, 21, 10, 12

Offset: 1

Omar E. Pol, Sep 03 2011

Sequence related to the D-toothpick "narrow" triangle (See A194442 and A194443). First differences of A194694.

			Written as a triangle:
2,
2,
4,
6,8,
8,11,16,13,
10,12,20,22,26,24,36,21,
10,12

Cf. A152980, A194271, A194442, A194443, A194693, A194694, A194697

a(n) = A194697(n)/2.

A160797 First differences of A160796.

Original entry on oeis.org

1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 3, 21, 9, 63, 9, 63, 27, 189, 9, 63, 27, 189, 27, 189, 81, 567, 3, 21, 9, 63, 9, 63, 27, 189, 9, 63, 27, 189, 27, 189, 81, 567, 9, 63, 27, 189, 27, 189, 81, 567, 27, 189, 81, 567, 81

Offset: 1

Omar E. Pol, Jun 13 2009

Cf. A152980, A160796.

More terms from Sean A. Irvine, Mar 22 2011

A161210 Toothpick sequence starting at the outside corner of an infinite cube from which protrudes a half toothpick.

Original entry on oeis.org

0, 1, 3, 7, 14, 21, 28

Offset: 0

Omar E. Pol, Jun 06 2009

The sequence gives the number of toothpicks after n rounds. A161211 (the first differences) gives the number added at the n-th round.
This structure is a three-dimensional version of the toothpick structure of A153006.
Toothpicks are placed following a rotation of axes: x,y,z,x,y,z,... and so on.
See A139250 and A160160 for more information about the toothpick sequences.

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, A152980, A153006, A160160, A161211, A161212, A161214, A161216.

cp's OEIS Frontend

A151694 G.f.: Product_{k>=1} (1 + 2*x^(2^k-1) + 2*x^(2^k)).

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Mathematica

A169708 First differences of A169707.

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Extensions

A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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Programs

Mathematica

Formula

Extensions

A153007 Triangular number A000217(n) minus toothpick number A153006(n).

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Author

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Crossrefs

Formula

Extensions

A159795 a(n) = 4*A153006(n).

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Extensions

A151555 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

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Author

Keywords

Comments

Examples

Links

Crossrefs

A194693 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194692.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A194695 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194694.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A160797 First differences of A160796.

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Author

Keywords

Crossrefs

Extensions

A161210 Toothpick sequence starting at the outside corner of an infinite cube from which protrudes a half toothpick.

A151694 G.f.: Product_{k>=1} (1 + 2x^(2^k-1) + 2x^(2^k)).