A172311 -id:A172311 - cp's OEIS frontend

A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.

0, 1, 2, 2, 2, 4, 4, 2, 4, 6, 6, 8, 6, 12, 10, 6, 4, 6, 10, 12, 10, 14, 14, 16, 16, 18, 14, 16, 12, 20, 22, 8, 8, 16, 18, 16, 10, 16, 20, 24, 26, 36, 24, 32, 30, 40, 28, 24

Offset: 0

Omar E. Pol, Feb 06 2010

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. First differences of A172308.

Cf. A139250, A139251, A152978, A172304, A172305, A172306, A172307, A172308, A172310, A172311, A172312, A172313.

a(0) = 0, a(n) = A172305(n+1)/4, for n>=1.

More terms from Colin Barker, Apr 19 2015

A172313 a(n) = A172309(n+1)/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 4, 3, 6, 5, 3, 2, 3, 5, 6, 5, 7, 7, 8, 8, 9, 7, 8, 6, 10, 11, 4, 4, 8, 9, 8, 5, 8, 10, 12, 13, 18, 12, 16, 15, 20, 14, 12

Offset: 1

Omar E. Pol, Feb 06 2010

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, A172304, A172305, A172306, A172307, A172308, A172309, A172310, A172311, A172312.

a(16)-a(46) from Robert Price, Jun 17 2019

A173531 a(0)=0: For n>0, a(n) = A060632(n)*A060632(n-1).

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 16, 8, 4, 8, 16, 16, 16, 32, 64, 16, 4, 8, 16, 16, 16, 32, 64, 32, 16, 32, 64, 64, 64, 128, 256, 32, 4, 8, 16, 16, 16, 32, 64, 32, 16, 32, 64, 64, 64, 128, 256, 64, 16, 32, 64, 64, 64, 128, 256, 128, 64, 128, 256, 256

Offset: 0

Omar E. Pol, Oct 10 2010

nonn

First differences of A173530.

Number of triangles (Or V-toothpicks, or L-toothpicks, etc.) added in the three-dimensional structure of A173530 at the n-th stage.

			If written as a triangle, begins:
0;
1;
2;
4,4;
4,8,16,8;
4,8,16,16,16,32,64,16;
4,8,16,16,16,32,64,32,16,32,64,64,64,128,256,32;
4,8,16,16,16,32,64,32,16,32,64,64,64,128,256,64,16,32,64,64,64,128,256,128,...

Michael De Vlieger, Table of n, a(n) for n = 0..16383
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.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
Michael De Vlieger, Scatterplot of Log_2 a(n) for n = 1..1024.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A001316, A047999, A060632, A139250, A139251, A161207, A172311, A173530, A173532.

Prepend[Times @@@ Partition[Array[2^DigitCount[Floor[#/2], 2, 1] &, 120, 0], 2, 1], 0] (* Michael De Vlieger, Jan 11 2024 *)

A173532 a(n) = A173531(n) - A139251(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 4, 0, 4, 32, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4, 28, 24, 4, 40, 176, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4, 28, 24, 4, 40, 176, 28, 0, 4, 28, 24, 4, 40, 172, 72, 4, 36, 144, 116, 48, 256, 832, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4

Offset: 0

Omar E. Pol, Oct 10 2010

nonn

			If written as a triangle, begins:
0;
0;
0,0;
0,0,0,4;
0,0,0,4,4,0,4,32;
0,0,0,4,4,0,4,32,12,0,4,28,24,4,40,176;
0,0,0,4,4,0,4,32,12,0,4,28,24,4,40,176,28,0,4,28,24,4,40,172,72,4,36,144,116,48,256,832;

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, A161207, A172311, A173530, A173531.

cp's OEIS Frontend

A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.

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A172313 a(n) = A172309(n+1)/2.

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A173531 a(0)=0: For n>0, a(n) = A060632(n)*A060632(n-1).

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A173532 a(n) = A173531(n) - A139251(n).

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