A153006 -id:A153006 - cp's OEIS frontend

A160796 Total number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton which is the "corner" structure corresponding to A160118.

0, 1, 8, 11, 32, 35, 56, 65, 128, 131, 152, 161, 224, 233, 296, 323, 512, 515, 536, 545, 608, 617, 680, 707, 896, 905, 968, 995, 1184, 1211, 1400, 1481, 2048, 2051, 2072, 2081, 2144, 2153, 2216, 2243, 2432, 2441, 2504, 2531, 2720, 2747, 2936, 3017, 3584, 3593, 3656

Offset: 0

Omar E. Pol, Jun 13 2009, Jun 14 2009

nonn

This bears the same relationship to A160118 as A153006 does to A139250.

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
..9...............9
...888.888.888.888.
...878.878.878.878.
...8866688.8866688.
.....656.....656...
...8866444.4446688.
...878.434.434.878.
...888.4422244.888.
.........212.......
00000000002244.888.
0000000000.434.878.
0000000000.4446688.
0000000000...656...
0000000000.8866688.
0000000000.878.878.
0000000000.888.888.
0000000000........9
0000000000.........
0000000000.........

Cf. A139250, A153006, A160797, A160798, A160412, A160118, A160416.

With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := (5 + 3 * If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]) / 4; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

a(n) = 2 + (3/4)*(A160118(n) - 1) if n >= 2.

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010
More terms from Nathaniel Johnston, Nov 13 2010
Corrected by Sean A. Irvine, Mar 23 2011, in response to correction to A160118
More terms from Amiram Eldar, Aug 01 2023

A194692 Corner sequence associated with A194440 and A194441.

Original entry on oeis.org

0, 2, 4, 8, 14, 22, 30, 42, 55, 67, 77, 93, 113, 145, 165, 189, 216, 236, 246, 262, 286, 322, 363, 411, 459, 513, 547, 591, 641, 721, 769, 817, 872, 908, 918, 934, 958, 994, 1035, 1083, 1137, 1205, 1267, 1347, 1427, 1551, 1646, 1750, 1839, 1933, 1977, 2025, 2093

Offset: 0

Omar E. Pol, Sep 01 2011

nonn

The sequence gives the number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "corner" structure related to the D-toothpick "wide" triangle (See A194440). The first differences (A194693) give the number of toothpicks or D-toothpicks added at n-th stage.

Cf. A153006, A170883, A194270, A194440, A194441, A194694.

A194694 Corner sequence associated with A194442 and A194443.

Original entry on oeis.org

0, 2, 4, 8, 14, 22, 30, 41, 57, 70, 80, 92, 112, 134, 160, 184, 220, 241, 251, 263

Offset: 0

Omar E. Pol, Sep 01 2011

The sequence gives the number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "corner" structure related to the D-toothpick "narrow" triangle (See A194442). The first differences (A194695) give the number of toothpicks or D-toothpicks added at n-th stage.

Cf. A153006, A170883, A194270, A194442, A194443, A194692

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.

A160416 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, 1, 8, 11, 32, 39, 80, 89, 146, 159

Offset: 0

Omar E. Pol, May 20 2009, Jun 14 2009

nonn

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
..9...9...9...9...9
...888.888.888.888.
...878.878.878.878.
...886668666866688.
..9..656.656.656..9
...886644464446688.
...878.434.434.878.
...886644222446688.
..9..656.212.656..9
000000000022446688.
0000000000.434.878.
000000000064446688.
000000000056.656..9
000000000066866688.
0000000000.878.878.
0000000000.888.888.
00000000009...9...9
0000000000.........
0000000000.........

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, A160117, A160118, A160410, A160412, A160414.

Cf. A160796, A161417. [From Omar E. Pol, Jun 14 2009]

A161212 a(n) = A161210(n)*2.

Original entry on oeis.org

0, 2, 6, 14, 28, 42, 56

Offset: 0

Omar E. Pol, Jun 06 2009

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, A161210, A161213, A161214, A161216.

A161214 a(n) = A161210(n)*3.

Original entry on oeis.org

0, 3, 9, 21, 42, 63, 84

Offset: 0

Omar E. Pol, Jun 06 2009

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, A161210, A161212, A161215, A161216.

A161216 a(n) = A161210(n)*4.

Original entry on oeis.org

0, 4, 12, 28, 56, 84, 112

Offset: 0

Omar E. Pol, Jun 06 2009

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, A161210, A161212, A161214, A161217.

A168112 Toothpick sequence starting with a straight line, with angle = Pi/4, from which protrudes a half toothpick.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 13, 19, 26, 32, 35, 41, 48, 56, 65, 81, 98, 108, 111, 117, 124, 132, 141, 157, 174, 186, 195, 211, 230, 252, 283, 329, 370, 388, 391, 397, 404, 412, 421, 437, 454, 466, 475, 491, 510, 532, 563, 609, 650, 670

Offset: 0

Omar E. Pol, Dec 07 2009

nonn

On the infinite square grid, we start at round 0 drawing a straight line, with angle = Pi/4, from which protrudes a half toothpick.
At round 1 we place an orthogonal toothpick centered at the end.
In each subsequent round, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
The sequence gives the number of toothpicks after n rounds.
See also A168113, the first differences.
For more information see A139250, which is the main entry for this sequence.

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, A153000, A153006, A160406, A168113, A168114.

a(n) = A160730(n)/2. [From Nathaniel Johnston, Mar 28 2011]

Terms after a(34) from Nathaniel Johnston, Mar 28 2011

A152979 Records in A152978.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 15, 22, 23, 28, 35, 52, 64, 65, 74, 91, 122, 168, 176, 177, 186, 239, 304, 412, 512, 513, 524, 527, 599, 782, 1020, 1336, 1488, 1489, 1500, 1575, 1980, 2584, 3376, 4160, 4161, 4172, 4174, 4247, 4275, 4653, 5010, 6544, 8544

Offset: 1

Omar E. Pol, Dec 23 2008

nonn

This sequence also gives the records in A152980. - Nathaniel Johnston, Apr 10 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. A039251, A152768, A152968, A152973, A152978, A152998, A153000, A153006. Toothpick sequence: A139250.

a(16)-a(51) from Nathaniel Johnston, Apr 10 2011

cp's OEIS Frontend

A160796 Total number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton which is the "corner" structure corresponding to A160118.

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A194692 Corner sequence associated with A194440 and A194441.

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A194694 Corner sequence associated with A194442 and A194443.

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

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A160416 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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A161212 a(n) = A161210(n)*2.

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A161214 a(n) = A161210(n)*3.

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A161216 a(n) = A161210(n)*4.

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A168112 Toothpick sequence starting with a straight line, with angle = Pi/4, from which protrudes a half toothpick.

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A152979 Records in A152978.

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