A139250 -id:A139250 - cp's OEIS frontend

A194810 Indices k such that A139250(k) = A000979(n).

2, 4, 8, 32, 64, 256, 512, 2048, 32768, 2097152, 1073741824, 549755813888, 1125899906842624, 9223372036854775808, 9671406556917033397649408, 39614081257132168796771975168, 633825300114114700748351602688

Offset: 1

Omar E. Pol, Oct 23 2011

nonn

Indices k such that the number of toothpicks in the toothpick structure of A139250 after k-th stage equals the n-th Wagstaff prime A000979. All terms of this sequence are powers of 2 (see formulas).

For a picture of the n-th Wagstaff prime as a toothpick structure see the Applegate link "A139250: the movie version", then enter N = a(n) and click "Update", for N = a(n) <= 32768 (due to the resolution of the movie).

			For n = 5 we have that a(5) = 64, then we can see that the number of toothpicks in the toothpick structure of A139250 after 64 stages is 2731 which coincides with the fifth Wagstaff prime, so we can write A139250(64) = A000979(5) = 2731. See the illustration in the Applegate-Pol-Sloane paper, figure 3: T(64) = 2731 toothpicks.

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.]
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, arXiv:1004.3036 [math.CO], 2010.
David Applegate, The movie version
C. Caldwell's The Top Twenty, Wagstaff.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Wikipedia, Wagstaff prime

Cf. A000978, A000979, A007583, A001045, A127936, A127958, A127962, A127965, A139250, A139253.

2^Reap[Do[If[PrimeQ[1+Sum[2^(2n-1), {n, m}]], Sow[m]], {m, 100}]][[2, 1]] (* Jean-François Alcover, Oct 06 2018 *)

a(n) = 2^A127936(n) = 2^(floor(A000978(n)/2)) = 2^(ceiling(log_4(A000979(n)))).
A139250(2^n) = A007583(n), n >= 0.
A139250(a(n)) = A000979(n).

More terms from Omar E. Pol, Mar 14 2012

A255263 Differences between the total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood and the total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 12, 20, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 4, 12, 20, 12, 36, 80, 68, 12, 36, 80, 84, 96, 208, 352, 196, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 4, 12, 20, 12, 36, 80, 68, 12, 36, 80

Offset: 1

Omar E. Pol, Feb 19 2015

It appears that the graph of A162795 lies between the graphs of A147562 and A169707.

It appears that a(n) = 0 if and only if n is a member of A048645.

			Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
0;
0;
0,0;
0,0,4,0;
0,0,4,0,4,12,20,0;
0,0,4,0,4,12,20,0,4,12,20,12,36,80,68,0;
0,0,4,0,4,12,20,0,4,12,20,12,36,80,68,0,4,12,20,12,36,80,68,12,36,80,84,96,208,352,196,0;
...
It appears that if k is a power of 2 then T(j,k) = 0.

Cf. A011782, A048645, A139250, A160164, A162796, A169707, A147562, A255166, A255264.

a(n) = A169707(n) - A162795(n).

A160165 a(n) = A000969(n) minus toothpick number A139250(n+1).

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 0, 2, 10, 15, 17, 21, 22, 12, 0, 5, 23, 39, 52, 66, 78, 79, 77, 89, 106, 112, 112, 109, 87, 39, 0, 10, 50, 87, 121, 157, 190, 212, 232, 265, 303, 331, 352, 370, 370, 343, 325, 353, 402, 440, 472, 501, 511, 495, 484, 502, 518, 503, 469, 409, 282

Offset: 0

Omar E. Pol, May 23 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. A000969, A139250, 153006, A153007, A159790, A160166.

a(n) = A000969(n) - A139250(n+1).

More terms from Jinyuan Wang, Mar 14 2020

A160424 Partial sums of A139250.

Original entry on oeis.org

0, 1, 4, 11, 22, 37, 60, 95, 138, 185, 240, 307, 386, 481, 604, 759, 930, 1105, 1288, 1483, 1690, 1913, 2164, 2447, 2750, 3069, 3416, 3799, 4222, 4705, 5276, 5927, 6610, 7297, 7992, 8699, 9418, 10153, 10916, 11711, 12526, 13357, 14216, 15111, 16046, 17041, 18124

Offset: 0

Omar E. Pol, May 23 2009

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..65535
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.

CoefficientList[Series[(x/((1 - x)^2 (1 + 2 x))) (1 + 2 x Product[1 + x^(2^k-1) + 2 x^(2^k), {k, 0, 20}]), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2014 *)

G.f.: (x/((1-x)^2*(1+2*x))) * (1+2*x*prod(k>=0, 1+x^(2^k-1)+2*x^(2^k))). - Vincenzo Librandi, Aug 22 2014

More terms from N. J. A. Sloane, May 17 2010

A160762 Convolved with the toothpick sequence A139250 = (2*n - 1): (1, 3, 5, 7, ...).

Original entry on oeis.org

1, 0, 2, -2, 2, 2, 2, -6, 2, 2, 2, -2, 6, 6, -2, -14, 2, 2, 2, -2, 6, 6, -2, -10, 6, 6, 2, 2

Offset: 1

Gary W. Adamson, May 25 2009

The sequence convolved with (1, 2, 2, 2, ...) = A139251, (1, 2, 4, 4, 4, 8, 12, ...); i.e., first difference row of the toothpick sequence A139250.

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.

A162625 Number of cells turned "ON" at n-th stage of cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250.

Original entry on oeis.org

3, 4, 6, 6, 8, 12, 14, 10, 8, 12, 16, 20, 28, 36, 34, 18, 8, 12, 16, 20, 28, 36, 36, 28, 28, 40, 52, 68, 92, 104, 82, 34, 8, 12, 16, 20, 28, 36, 36, 28, 28, 40, 52, 68, 92, 104, 84, 44, 28, 40, 52, 68, 92, 108, 100, 84, 96, 132, 172, 228, 288, 288, 194, 66, 8

Offset: 1

Omar E. Pol, Aug 05 2009

nonn

First differences of A147614.

The main entry for this sequence is A139250, the toothpick sequence. See also A139251.

Jinyuan Wang, Table of n, a(n) for n = 1..1000
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, A147614, A160420, A160421, A160422, A160423, A162627.

More terms from Nathaniel Johnston, Nov 15 2010

More terms from Jinyuan Wang, Mar 03 2020

A194800 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 2 and the horizontal toothpicks have length 4.

Original entry on oeis.org

0, 3, 11, 17, 31, 39, 67

Offset: 0

Omar E. Pol, Sep 07 2011

There are an infinite family of these sequences since A139250 gives the number of toothpicks in the structure regardless of the length difference between horizontal toothpicks and vertical toothpicks. Examples: A147614, this sequence, A194802, A160420, etc.

			a(2) = 11.
o o o o o
. . o . .
o o o o o

Cf. A139250, A147614, A162795, A162796, A162797, A175098, A160420, A160422, A194802.

A194802 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 4 and the horizontal toothpicks have length 2.

Original entry on oeis.org

0, 5, 9, 23, 29, 45, 57

Offset: 0

Omar E. Pol, Sep 07 2011

There are an infinite family of these sequences since A139250 gives the number of toothpicks in the structure regardless of the length difference between horizontal toothpicks and vertical toothpicks. Examples: A147614, A194800, this sequence, A160420, etc.

			a(2) = 9.
o o o
. o .
. o .
. o .
o o o

Cf. A139250, A147614, A162795, A162796, A162797, A175098, A160420, A160422, A194800

A351837 Consider a variant of the toothpick sequence (A139250) where each new toothpick, except the first, touches exactly one existing toothpick at the ends, this one being in the prior stage; a(n) is the total number of toothpicks at stage n.

Original entry on oeis.org

0, 1, 5, 9, 17, 25, 37, 53, 69, 77, 89, 109, 133, 161, 201, 249, 281, 289, 301, 321, 345, 373, 413, 465, 505, 533, 577, 641, 717, 813, 941, 1069, 1133, 1141, 1153, 1173, 1197, 1225, 1265, 1317, 1357, 1385, 1429, 1493, 1569, 1665, 1793, 1925, 1997, 2025, 2069

Offset: 0

Rémy Sigrist, Feb 21 2022

nonn

We consider toothpicks of length 1, parallel to the X and Y axes.
We start at stage 0 with no toothpicks.
At stage 1 we place one toothpick anywhere in the plane.
At stage n > 1, we consider all exposed ends E (i.e. in contact with no other toothpick) and attach perpendicular toothpicks in contact with E by one end provided that they won't touch other existing toothpicks (from stages 1 to n-1).
A toothpick added at stage n may touch other toothpicks added at stage n.

			The configuration at stage 4 can be depicted as follows (stars representing ends and toothpicks being labeled with their stage of appearance):
.
         *                       *
         |                       |
         4                       4
         |                       |
         *---3---*       *---3---*
         |       |       |       |
         4       2       2       4
         |       |       |       |
         *       *---1---*       *
         |       |       |       |
         4       2       2       4
         |       |       |       |
         *---3---*       *---3---*
         |                       |
         4                       4
         |                       |
         *                       *
.
- so a(4) = 1 + 4 + 4 + 8 = 17.

Rémy Sigrist, Table of n, a(n) for n = 0..8194
Rémy Sigrist, Illustration of the structure at stage 16
Rémy Sigrist, PARI program
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A139250, A351838.

PARI
```
See Links section.
```

A139255 Complement of toothpick sequence A139250.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72

Offset: 1

Omar E. Pol, May 11 2008, Jan 02 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, A139252, A139253, A139254.

cp's OEIS Frontend

A194810 Indices k such that A139250(k) = A000979(n).

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A255263 Differences between the total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood and the total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A160165 a(n) = A000969(n) minus toothpick number A139250(n+1).

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Author

Keywords

Links

Crossrefs

Formula

Extensions

A160424 Partial sums of A139250.

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Author

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Programs

Mathematica

Formula

Extensions

A160762 Convolved with the toothpick sequence A139250 = (2*n - 1): (1, 3, 5, 7, ...).

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Author

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Comments

Links

Crossrefs

A162625 Number of cells turned "ON" at n-th stage of cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A194800 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 2 and the horizontal toothpicks have length 4.

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Author

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Comments

Examples

Links

Crossrefs

A194802 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 4 and the horizontal toothpicks have length 2.

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Author

Keywords

Comments

Examples

Links

Crossrefs

A351837 Consider a variant of the toothpick sequence (A139250) where each new toothpick, except the first, touches exactly one existing toothpick at the ends, this one being in the prior stage; a(n) is the total number of toothpicks at stage n.

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Author

Keywords

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Examples

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Programs