A160406 -id:A160406 - cp's OEIS frontend

A151922 Number of first-quadrant cells (including the two boundaries) that are "ON" after n-th stage of the Holladay-Ulam cellular automaton.

1, 3, 5, 9, 11, 15, 19, 29, 31, 35, 39, 49, 53, 63, 73, 101, 103, 107, 111, 121, 125, 135, 145, 173, 177, 187, 197, 225, 235, 263, 291, 373, 375, 379, 383, 393, 397, 407, 417, 445, 449, 459, 469, 497, 507, 535, 563, 645

Offset: 0

N. J. A. Sloane, Aug 05 2009

See A079314, the first differences and A147562, the main entry for this CA, for further information.

Paolo Xausa, Table of n, a(n) for n = 0..10000
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
Index entries for sequences related to cellular automata

Cf. A079314, A079316, A139250, A147562, A160406.

A151922list[nmax_]:=Accumulate[Join[{1},3^(DigitCount[Range[nmax],2,1]-1)+1]];A151922list[100] (* Paolo Xausa, Jun 29 2023 *)

a(n) = A079316(2n).

Extended (a(30)-a(47)) and edited by Omar E. Pol, Nov 21 2010

A160158 Toothpick sequence starting from a segment of length 4 formed by two toothpicks.

Original entry on oeis.org

0, 2, 4, 8, 16, 24, 28, 36, 48, 56, 64, 80, 104, 124, 136, 156, 176, 184, 192, 208, 232, 252, 268, 296, 328, 352, 384, 440, 508, 560, 600, 652, 688, 696, 704, 720, 744, 764, 780, 808, 840, 864, 896, 952, 1020, 1072, 1116, 1176, 1224, 1248, 1280, 1336, 1404

Offset: 0

Omar E. Pol, May 29 2009

nonn

The sequence gives the number of toothpicks in the sieve after n-th stage.

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, A160406, A160730, A160732, A160736, A160738, A160740.

Extended beyond a(13) by R. J. Mathar, Jun 01 2009

A160409 First differences of toothpick numbers A160408.

Original entry on oeis.org

1, 1, 2, 4, 4, 4, 4, 4, 8, 16, 16, 8, 4, 4, 8

Offset: 1

Omar E. Pol, May 23 2009

Number of toothpick added to the toothpick pyramid at the round n.

See also the toothpick sequences A139250, A160160 and the toothpick triangle A160406.

			Contribution from _Omar E. Pol_, Jun 06 2009: (Start)
Array begins:
========
x, y, z
========
1, 1, 2;
4, 4, 4;
4, 4, 8;
16, 16, 8;
4, 4, 8;
(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

Cf. A139250, A139251, A160160, A160161, A160406, A160407, A160408.

More terms from Omar E. Pol, Jun 06 2009

A160162 a(n) = A160158(n)/2.

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 14, 18, 24, 28, 32, 40, 52, 62, 68, 78, 88, 92, 96, 104, 116, 126, 134, 148, 164, 176, 192, 220, 254, 280, 300, 326, 344, 348, 352, 360, 372, 382, 390, 404, 420, 432, 448, 476, 510, 536, 558, 588, 612, 624, 640, 668, 702, 730, 760, 804, 848

Offset: 0

Omar E. Pol, May 29 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

Cf. A139250, A160158, A160406, A160730, A160732, A160736, A160738, A160740.

Extended beyond a(13) by R. J. Mathar, Jun 01 2009

A161412 V-toothpick sequence starting at the corner of an infinite square from which protrudes a half toothpick with an angle = Pi/6.

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 12, 15, 19, 25, 29, 34, 39, 49, 58, 63

Offset: 0

Omar E. Pol, Jun 10 2009

The sequence gives the number of V-toothpicks in the structure after n rounds. A161413 (the first differences) gives the number added at the n-th round.

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

V-toothpick sequence: A161206.

Cf. A139250, A160120, A160406, A161328, A161413.

A161420 V-toothpick sequence starting from a V-toothpick whose central point touch a straight line, as a V.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 19, 25, 31, 39, 51, 59, 69, 79, 99, 117, 127, 143, 163, 171

Offset: 0

Omar E. Pol, Jun 10 2009

The sequence gives the number of V-toothpicks after n-th stages on hexagonal net. A161421 (the first differences) gives the number added at the n-th stage. See A161206 for more information.

Also, it appears this is a H-toothpick sequence in the first quadrant on the square grid, starting with a D-toothpick from the point (0,0). The sequence gives the number of toothpicks and D-toothpicks after n-th stage. A161421 (the first differences) gives the number added at the n-th stage. For more information see A182838.

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, A160120, A160406, A161206, A161328, A161412, A161421, A161422, A161423, A182832, A182838.

A161832 a(n) = (A161830(n+1)-1)/2.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 13, 15, 17, 19, 23, 26, 30, 35, 44, 49, 51

Offset: 0

Omar E. Pol, Jun 20 2009

The sequence gives the number of Y-toothpicks in the structure after n rounds. A161833 (the first differences) gives the number added at the n-th round.

See the Y-toothpick sequence A160120 and A161830 for more information.

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, A160120, A160406, A161830, A161831, A161833.

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

A160418 a(n) = A160407(n+2)/2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 5, 4, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 2, 3, 5, 5, 6, 10, 13, 9, 6, 10, 14, 15, 21, 32, 33, 16, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 2, 3, 5, 5

Offset: 1

Omar E. Pol, May 23 2009

Row lengths are the terms of A000079 multiplied by 2. Right border gives A000079. - Omar E. Pol, Mar 19 2020

			From _Omar E. Pol_, Mar 19 2020: (Start)
Triangle begins:
  1,1;
  1,1,2,2;
  1,1,2,2,2,3,5,4;
  1,1,2,2,2,3,5,4,2,3,5,5,6,10,13,8;
  1,1,2,2,2,3,5,4,2,3,5,5,6,10,13,8,2,3,5,5,6,10,13,9,6,10,14,15,21,32,33,16;
  ... (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

Cf. A139250, A139251, A160160, A160161, A160406, A160407, A160408, A160409.

More terms from Jinyuan Wang, Mar 14 2020

A160728 Toothpick cube: a(n) = A160408(n)*6.

Original entry on oeis.org

0, 6, 12, 24, 48, 72, 96, 120, 144, 192, 288, 384, 432, 456, 480, 528

Offset: 0

Omar E. Pol, Jun 06 2009

Also, 6 times the toothpick pyramid A160408.
The sequence gives the number of toothpicks after n rounds. A160729 (the first differences) gives the number added at the n-th round.
See also the entry A139250 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

Toothpick triangle: A160406. Toothpick pyramid: A160408.

Cf. A139250, A139251, A160160, A160729.

cp's OEIS Frontend

A151922 Number of first-quadrant cells (including the two boundaries) that are "ON" after n-th stage of the Holladay-Ulam cellular automaton.

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A160158 Toothpick sequence starting from a segment of length 4 formed by two toothpicks.

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A160409 First differences of toothpick numbers A160408.

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A160162 a(n) = A160158(n)/2.

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A161412 V-toothpick sequence starting at the corner of an infinite square from which protrudes a half toothpick with an angle = Pi/6.

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A161420 V-toothpick sequence starting from a V-toothpick whose central point touch a straight line, as a V.

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A161832 a(n) = (A161830(n+1)-1)/2.

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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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A160418 a(n) = A160407(n+2)/2.

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A160728 Toothpick cube: a(n) = A160408(n)*6.

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