A152968 -id:A152968 - cp's OEIS frontend

A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

0, 1, 6, 16, 26, 36, 56, 86, 106, 116, 136, 176, 216, 246, 296, 366, 406, 416, 436, 476, 536, 616

Offset: 0

Omar E. Pol, Mar 15 2013

Analog of A161644, A147562 and A151723, but here we are working without a lattice. Each regular pentagon has five virtual neighbors. Overlapping are prohibited. The sequence gives the number of pentagons in the structure after n-th stage. A222181 (the first differences) gives the number of pentagons added at n-th stage.

Also this is a P-toothpick sequence since every pentagon can be replaced by a P-toothpick which is formed by five toothpicks as a five-pointed star. Note that each toothpick can be represented as an apothem or as a radius of a pentagon. In both types of structures the number of toothpicks after n-th stage is equal to 5*a(n).

Omar E. Pol, Illustration of structure after 5 stages, (Contains 36 pentagons).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A147562, A151723, A152968, A161330, A161644, A182632, A182840, A222172, A222176, A222181.

a(n) = 6 + 10*A222172(n-2), n >= 2. - Omar E. Pol, Nov 24 2013

Name improved by Omar E. Pol, Nov 24 2013

A222181 Number of pentagons added at n-th stage to the structure of A222180.

Original entry on oeis.org

0, 1, 5, 10, 10, 10, 20, 30, 20, 10, 20, 40, 40, 30, 50, 70, 40, 10, 20, 40, 60, 80

Offset: 0

Omar E. Pol, Mar 15 2013

Essentially the first differences of A222180.

Also number of P-toothpicks added at n-th stage to the P-toothpick structure of A222180.

			Apparently this is an irregular triangle:
0;
1;
5;
10,10;
10,20,30,20;
10,20,40,40,30,50,70,40;
10,20,40,60,80,...

Cf. A139250, A139251, A147582, A151724, A152968, A161645, A182633, A182841, A222173, A222177, A222180.

a(n) = 10*A222173(n-2), n >= 3. - Omar E. Pol, Nov 24 2013

A222177 Number of pentagons added at n-th stage to the structure of A222176.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 6, 4, 2, 4, 8, 8, 6, 10, 14, 8, 2, 4, 8, 12, 16

Offset: 0

Omar E. Pol, Mar 21 2013

Essentially the first differences of A222176.
Also number of P-toothpicks added at n-th stage to the P-toothpick structure of A222176.
First differs from A152968 at a(10).

			Apparently this is an irregular triangle:
0;
1;
2, 2;
2, 4, 6, 4;
2, 4, 8, 8, 6, 10, 14, 8;
2, 4, 8, 12, 16,...

Cf. A139251, A152968, A222173, A222176, A222180, A222181.

a(n) = A222181(n+1)/5, n >= 1.

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

A152999 Primes in toothpick sequence A152998.

Original entry on oeis.org

3, 5, 7, 11, 17, 23, 47, 61, 97, 103, 151, 173, 191, 211, 241, 347, 353, 359, 367, 397, 467, 541, 599, 607, 659, 733, 1109, 1237, 1367, 1439, 1453, 1471, 1663, 2029, 2357, 2399, 2671, 2797, 3373, 3607, 3719, 3911, 4241, 5479, 5501, 5527, 5701, 5741, 5779, 5923

Offset: 1

Omar E. Pol, Dec 23 2008

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, A139253, A152768, A152968, A152978, A152998, A153000, A153002, A153006, A153009.

More terms from Jinyuan Wang, Mar 15 2020

A159785 a(n) = A152980(n)*3.

Original entry on oeis.org

3, 6, 9, 9, 12, 21, 24, 15, 12, 21, 27, 30, 45, 66, 60, 27, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 144, 51, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 147, 66, 45, 69, 84, 105, 156, 195, 168, 129, 159, 222, 273, 366, 504, 528

Offset: 1

Omar E. Pol, May 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

Toothpick sequence: A139250.

Cf. A139251, A147646, A152968, A152978, A152980, A153004, A153006.

More terms from 3*A152980(n) by Jinyuan Wang, Mar 14 2020

A162627 a(n) = A162625(n+1)/2.

Original entry on oeis.org

2, 3, 3, 4, 6, 7, 5, 4, 6, 8, 10, 14, 18, 17, 9, 4, 6, 8, 10, 14, 18, 18, 14, 14, 20, 26, 34, 46, 52, 41, 17, 4, 6, 8, 10, 14, 18, 18, 14, 14, 20, 26, 34, 46, 52, 42, 22, 14, 20, 26, 34, 46, 54, 50, 42, 48, 66, 86, 114, 144, 144, 97, 33, 4, 6, 8, 10, 14, 18, 18

Offset: 1

Omar E. Pol, Aug 05 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, A147614, A152968, A160421, A160423, A162625.

More terms from Jinyuan Wang, Mar 03 2020

cp's OEIS Frontend

A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A222181 Number of pentagons added at n-th stage to the structure of A222180.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A222177 Number of pentagons added at n-th stage to the structure of A222176.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A152979 Records in A152978.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A152999 Primes in toothpick sequence A152998.

Views

Author

Keywords

Links

Crossrefs

Extensions

A159785 a(n) = A152980(n)*3.

Views

Author

Keywords

Links

Crossrefs

Extensions

A162627 a(n) = A162625(n+1)/2.

Views

Author

Keywords

Links

Crossrefs

Extensions