A160426 -id:A160426 - cp's OEIS frontend

A160427 First differences of A160426.

5, 4, 8, 13, 12, 10, 17, 21, 12, 10, 17, 21, 20, 26, 41, 37, 12, 10, 17, 21, 20, 26, 41, 37, 20, 26, 41, 45, 52, 82, 105, 69, 12, 10, 17, 21, 20, 26, 41, 37, 20, 26, 41, 45, 52, 82, 105, 69, 20, 26, 41, 45, 52, 82, 105

Offset: 1

Omar E. Pol, May 25 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, A139251.

Terms after a(13) from Nathaniel Johnston, Mar 31 2011

A160172 T-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468

Offset: 0

Omar E. Pol, Jun 01 2009

A T-toothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no T-toothpicks.
At round 1 we place a T-toothpick anywhere in the plane.
At round 2 we place three other T-toothpicks.
And so on...
The rule for adding a new T-toothpick is the following. A new T-toothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.
If the T-toothpick has three components then at the n-th round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every T-toothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section. - Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

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

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

a(n) = 2*A151920(n) + 2*A151920(n-1) + n + 1. - Charlie Neder, Feb 07 2019

Edited and extended by N. J. A. Sloane, Jan 01 2010

A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 8, 16, 24, 32, 40, 56, 72, 80, 88, 104, 120, 136, 160, 200, 232, 240, 248, 264, 280, 296, 320, 360, 392, 408, 432, 472, 512, 560, 640, 744, 808, 816, 824, 840, 856, 872, 896, 936, 968, 984, 1008, 1048, 1088, 1136, 1216, 1320, 1384, 1400, 1424, 1464, 1504, 1552

Offset: 0

Omar E. Pol, May 25 2009

nonn

On the infinite square grid we start at stage 0 with no toothpicks. Toothpicks have length 2. At stage 1 we place two consecutive toothpicks in the vertical direction and two consecutive toothpicks in the horizontal direction forming a cross centered at the origin. At stage 2 we place four toothpicks. At stage 3 we place eight toothpicks. For more information about the toothpick sequences see A139250. - Omar E. Pol, Nov 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.], which is also available at arXiv:1004.3036v2
Mathematical Association of America, NumberADay: 1136
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160736, A160738, A160741.

Cf. A160170, A160426, A194432, A194434.

a(n) = 4*A160406(n).

More terms from N. J. A. Sloane, May 25 2009

A160800 Toothpick sequence in Fibonacci spiral (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 9, 10, 12, 15, 16, 17, 19, 22, 25, 28, 33, 38, 40, 41, 43, 46, 48, 50, 54, 60, 65, 68, 73, 78, 80, 82, 86, 92, 93, 95, 98, 100, 102, 106, 112, 117, 120, 125, 130, 132, 134, 138, 144, 149, 153, 160, 168, 174, 182, 196

Offset: 0

Omar E. Pol, May 26 2009

nonn

On the infinite square grid we draw a Fibonacci spiral starting with 2,2,4,6,10,16,... (Note that each edge has length = A000045(k)*2, for k>0). We start at stage 0 with no toothpicks. At stage 1 we place a toothpick in a orthogonal direction, in the center of the Fibonacci spiral. At stage 2 we place 2 toothpicks. And so on...

The sequence gives the number of toothpicks in the Fibonacci spiral after n stages. A160801 (the first differences) gives the number added at the n-th stage. See A139250 for more information about toothpick sequences.

Nathaniel Johnston, Table of n, a(n) for n = 0..255
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
Nathaniel Johnston, C script for computing terms

Cf. A000045, A139250, A139251, A160426, A160427.

Terms after a(32) from Nathaniel Johnston, Mar 30 2011

A160801 First differences of A160800.

Original entry on oeis.org

1, 2, 4, 2, 1, 2, 3, 1, 1, 2, 3, 3, 3, 5, 5, 2, 1, 2, 3, 2, 2, 4, 6, 5, 3, 5, 5, 2, 2, 4, 6, 1, 2, 3, 2, 2, 4, 6, 5, 3, 5, 5, 2, 2, 4, 6, 5, 4, 7, 8, 6, 8, 14, 17, 13, 10, 9, 8, 3, 3, 4, 5, 6, 7, 8, 7, 6, 8, 11, 14, 17, 19, 10

Offset: 1

Omar E. Pol, May 26 2009

nonn

Number of toothpicks added to the spiral at the 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. A000045, A139250, A139251, A160426, A160427, A160800.

Terms after a(32) from Nathaniel Johnston, Mar 30 2011

A160802 Toothpick sequence in Fibonacci spiral (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 25, 28, 30, 33, 37, 38, 40, 43, 45, 47, 51, 57, 62, 65, 70, 75, 77, 79, 81, 83, 86, 90, 93, 95, 98, 102, 106, 111, 119, 127, 131, 132, 134, 137, 140, 144, 151, 159, 164, 168, 175, 184, 194, 209, 231, 243, 250

Offset: 0

Omar E. Pol, May 26 2009

nonn

On the infinite square grid we draw a Fibonacci spiral starting with 3,3,6,9,15,24,... (Note that each edge has length = A000045(k)*3, for k>0). We start at stage 0 with no toothpicks. At stage 1 we place a toothpick of length 2 in a orthogonal direction, in the center of the Fibonacci spiral. At stage 2 we place 2 toothpicks. And so on...

The sequence gives the number of toothpicks in the Fibonacci spiral after n stages. A160803 (the first differences) gives the number added at the n-th stage. See A160800 and A139250 for more information about toothpick sequences.

Nathaniel Johnston, Table of n, a(n) for n = 0..256
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
Nathaniel Johnston, C script for computing terms

Cf. A000045, A139250, A139251, A160426, A160427, A160800, A160801, A160803.

Terms after a(12) from Nathaniel Johnston, Mar 30 2011

A160803 First differences of A160802.

Original entry on oeis.org

1, 2, 4, 4, 4, 6, 4, 3, 2, 3, 4, 1, 2, 3, 2, 2, 4, 6, 5, 3, 5, 5, 2, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 8, 8, 4, 1, 2, 3, 3, 4, 7, 8, 5, 4, 7, 9, 10, 15, 22, 12, 7, 2, 2, 3, 4, 5, 6, 7, 6, 5, 7, 10, 13, 16, 18, 9, 8, 3, 3, 6, 9, 8, 7

Offset: 1

Omar E. Pol, May 26 2009

nonn

Number of toothpicks added to the spiral at the n-th stage.

Nathaniel Johnston, Table of n, a(n) for n = 1..254
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. A000045, A139250, A139251, A160426, A160427, A160800, A160801, A160802.

Terms after a(12) from Nathaniel Johnston, Mar 30 2011

A160808 Toothpick sequence in Fibonacci spiral (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 35, 39, 40, 42, 45, 48, 52, 59, 67, 68, 70, 73, 76, 80, 85, 92, 97, 100, 105, 112, 120, 131, 144, 161, 173, 177, 182, 190, 197, 206, 211, 218, 227, 235, 239, 247, 255, 262, 270, 283, 297

Offset: 0

Omar E. Pol, May 26 2009

nonn

On the infinite square grid we draw a Fibonacci spiral starting with 4,4,8,12,20,32,... (Note that each edge has length = A000045(k)*4, for k>0). We start at stage 0 with no toothpicks. At stage 1 we place a toothpick of length 2 in a orthogonal direction, in the center of the Fibonacci spiral. At stage 2 we place 2 toothpicks. And so on... The sequence gives the number of toothpicks in the Fibonacci spiral after n stages. A160809 (the first differences) gives the number added at the n-th stage. See 160800, A160802 and A139250 for more information about toothpick sequences.

Nathaniel Johnston, Table of n, a(n) for n = 0..256
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
Nathaniel Johnston, C script for computing terms

Cf. A000045, A139250, A139251, A160426, A160427, A160800, A160801, A160802, A160803, A160809.

Terms after a(16) from Nathaniel Johnston, Mar 30 2011

A160809 First differences of A160808.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 12, 4, 1, 2, 3, 3, 4, 7, 8, 1, 2, 3, 3, 4, 5, 7, 5, 3, 5, 7, 8, 11, 13, 17, 12, 4, 5, 8, 7, 9, 5, 7, 9, 8, 4, 8, 8, 7, 8, 13, 14, 10, 8, 12, 11, 13, 13, 17, 16, 11, 8, 15, 13, 14, 19, 26, 21, 5, 7

Offset: 1

Omar E. Pol, May 26 2009

nonn

Number of toothpicks added to the spiral at the n-th stage.

Nathaniel Johnston, Table of n, a(n) for n = 1..254
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. A000045, A139250, A139251, A160426, A160427, A160800, A160801, A160802, A160803, A160808.

Terms after a(16) from Nathaniel Johnston, Mar 30 2011

cp's OEIS Frontend

A160427 First differences of A160426.

Views

Author

Keywords

Links

Crossrefs

Extensions

A160172 T-toothpick sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A160800 Toothpick sequence in Fibonacci spiral (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160801 First differences of A160800.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160802 Toothpick sequence in Fibonacci spiral (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160803 First differences of A160802.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160808 Toothpick sequence in Fibonacci spiral (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160809 First differences of A160808.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions