A153000 -id:A153000 - cp's OEIS frontend

A222172 Total number of ON states after n generations of cellular automaton based on pentagons starting near of the vertex of an infinite 36-degree wedge.

0, 1, 2, 3, 5, 8, 10, 11, 13, 17, 21, 24, 29, 36, 40, 41, 43, 47, 53, 61

Offset: 0

Omar E. Pol, Mar 21 2013

The sequence gives the number of pentagons in the structure after n-th stage. A222173 (the first differences) gives the number of pentagons added at n-th stage.
Also P-toothpick sequence starting near of the vertex of an infinite 36-degree wedge, since each pentagon can be replaced by a P-toothpick.
For more information see A222180.
First differs from A153000 at a(9).

Cf. A139250, A153000, A222173, A222176, A222180.

a(n) = (((A222180(n+2) - 1)/5) - 1)/2 = (A222176(n+1) - 1)/2.

a(n) = (A222180(n+2) - 6)/10. - Omar E. Pol, Nov 24 2013

Name improved by Omar E. Pol, Nov 24 2013

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

A159799 First differences of A159789.

Original entry on oeis.org

0, 1, 1, 0, 1, 5, 4, 0, 1, 3, 2, 1, 7, 18, 12, 0, 1, 3, 2, 1, 7, 14, 8, 1, 5, 8, 5, 9, 32, 64, 24, 0, 1, 3, 2, 1, 7, 14, 8, 1, 5, 8, 5, 9, 32, 48, 24, 1, 5

Offset: 1

Omar E. Pol, Jun 13 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, A153000, A159786, A159789, A159796.

a(n) = A159796(n+1)/4.

More terms from Colin Barker, Apr 19 2015

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

A160718 a(n) = A160406(n+2)/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 17, 20, 25, 29, 30, 31, 33, 35, 37, 40, 45, 49, 51, 54, 59, 64, 70, 80, 93, 101, 102, 103, 105, 107, 109, 112, 117, 121, 123, 126, 131, 136, 142, 152, 165, 173, 175, 178, 183, 188, 194, 204, 217, 226, 232, 242, 256, 271, 292

Offset: 0

Omar E. Pol, Jun 10 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, A152998, A153000, A160406, A160407, A160719.

a(n) = A160719(n) + 1 = -a(n-1) + n + (A139250(n+1) + 5)/4 for n > 0. - Jinyuan Wang, Mar 04 2020

More terms from Jinyuan Wang, Mar 04 2020

A160719 a(n) = A160406(n+2)/2 - 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 19, 24, 28, 29, 30, 32, 34, 36, 39, 44, 48, 50, 53, 58, 63, 69, 79, 92, 100, 101, 102, 104, 106, 108, 111, 116, 120, 122, 125, 130, 135, 141, 151, 164, 172, 174, 177, 182, 187, 193, 203, 216, 225, 231, 241, 255, 270, 291

Offset: 0

Omar E. Pol, Jun 10 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, A153000, A160406, A160407, A160718.

More terms from Jinyuan Wang, Mar 04 2020

A160734 a(n) = (A160158(n+2)-4)/4.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 11, 13, 15, 19, 25, 30, 33, 38, 43, 45, 47, 51

Offset: 0

Omar E. Pol, Jun 10 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, A153000, A160158, A160162, A160735.

A153005 Primes in toothpick sequence A153003.

Original entry on oeis.org

7, 31, 127, 211, 487, 571, 643, 811, 1033, 1249, 1663, 1999, 2131, 2179, 2281, 2347, 2467, 3391, 4801, 5059, 6361, 7759, 8191, 8209, 8713, 8779, 8929, 9187, 9343, 9679, 9931, 10687, 13903, 14947, 19009, 19267, 19423, 25057, 26731, 28879, 33289, 35521

Offset: 1

Omar E. Pol, 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, A139253, A152998, A152999, A153000, A153002, A153003, A153006, A153009.

read("transforms3") ; a139250 := BFILETOLIST("b139250.txt") ; A139250 := proc(n) global a139250; op(n+1,a139250) ; end: A153003 := proc(n) if n =0 then 0; else 1+3/4*(A139250(n+1)-3) ; fi; end: for n from 0 to 400 do p := A153003(n) ; if isprime(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 13 2009

a139250 = Cases[Import["https://oeis.org/A139250/b139250.txt", "Table"], {, }][[All, 2]];
A139250[n_] := a139250[[n + 1]];
A153003[n_] := If[n == 0, 0, 1 + 3/4*(A139250[n + 1] - 3)];
Reap[Do[p = A153003[n]; If[PrimeQ[p], Sow[p]], {n, 0, 400}]][[2, 1]] (* Jean-François Alcover, Apr 05 2020 *)

More terms from R. J. Mathar, Jul 13 2009

A153838 Records in A153004.

Original entry on oeis.org

1, 3, 6, 9, 12, 21, 24, 27, 30, 45, 66, 69, 84, 105, 156, 192, 195, 222, 273, 366, 504, 528, 531, 558, 717, 912, 1236, 1536, 1539, 1572, 1581, 1797, 2346, 3060, 4008, 4464, 4467, 4500, 4725, 5940, 7752, 10128, 12480, 12483, 12516, 12522, 12741, 12825, 13959

Offset: 1

Omar E. Pol, 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. A139251, A152768, A152973, A152978, A152979, A152998, A153000, A153003, A153004. Toothpick sequence: A139250.

read("transforms3") : a139250 := BFILETOLIST("b139250.txt") : A139250 := proc(n) global a139250; op(n+1,a139250) ; end: A153003 := proc(n) if n =0 then 0; else 1+3/4*(A139250(n+1)-3) ; fi; end: A153004 := proc(n) A153003(n)-A153003(n-1) ; end: rec := -1 ; for n from 1 to 3000 do t := A153004(n) ; if t > rec then printf("%d,",t) ; rec := t; fi; od: # R. J. Mathar, Jul 13 2009

More terms from R. J. Mathar, Jul 13 2009

A168002 Duplicate of A079559.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1

Offset: 1

Omar E. Pol, Nov 29 2009

dead

Name was: Parity of positive integers of the toothpick sequence in the first quadrant A153000.

Same as A079559: reduce the g.f. of A153000 mod 2 to see this. [Joerg Arndt, Aug 07 2013]

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. A040051, A079559, A139250, A153000

a(n) = A000035(A153000(n)). - Omar E. Pol, Aug 06 2013

More terms from R. J. Mathar, Dec 04 2009

cp's OEIS Frontend

A222172 Total number of ON states after n generations of cellular automaton based on pentagons starting near of the vertex of an infinite 36-degree wedge.

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A152979 Records in A152978.

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A159799 First differences of A159789.

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A152999 Primes in toothpick sequence A152998.

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A160718 a(n) = A160406(n+2)/2.

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A160719 a(n) = A160406(n+2)/2 - 1.

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A160734 a(n) = (A160158(n+2)-4)/4.

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A153005 Primes in toothpick sequence A153003.

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A153838 Records in A153004.

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Maple

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A168002 Duplicate of A079559.

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