A161331 -id:A161331 - cp's OEIS frontend

A161332 a(n) = A161331(n)/2.

0, 1, 3, 3, 3, 9, 3, 9, 9, 9, 15, 9, 15, 21, 3, 9, 9, 15, 27, 15, 39, 21, 27, 39, 21, 33, 39, 9, 21, 21, 27, 45, 33, 63, 45, 45, 51, 33, 39, 45, 45, 45, 27, 33, 57, 39, 63, 63, 51, 51, 69, 51, 81, 51, 57, 81, 63, 81, 57, 51, 51, 63, 93, 93, 75, 69, 63

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

Number of E-toothpicks added at n-th stage to the structure of A161334. - Omar E. Pol, Jan 07 2014

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, A single E-toothpick ç
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A161330, A161331, A161334.

More terms from Omar E. Pol, Jan 07 2014

A160121 First differences of A160120.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 9, 9, 9, 21, 15, 21, 27, 51, 27, 9, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 81, 21, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 87, 33, 27, 51, 51, 57, 75, 129, 117, 75, 69, 117, 135, 141, 171, 279, 231, 69, 9, 21, 15, 21, 27

Offset: 1

Omar E. Pol, May 02 2009

nonn

Number of Y-toothpicks added at n-th stage to the Y-toothpick structure of A160120.

For a simpler version, see A151710. - Omar E. Pol, Dec 18 2012

			Contribution from _Omar E. Pol_, Jun 18 2009: (Start)
May be written as a triangle:
1,
3,
3,
9,
3,9,
9,21,9,9,
9,21,15,21,27,51,27,9,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,81,21,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279,231,69;
Rows converge to A161326.
(End)
Contribution from _Omar E. Pol_, Dec 18 2012: (Start):
Also this sequence may be written as another triangle (according to the structure of triangle A151710):
1;
3;
3,  9;
3,  9,9,21;
9,  9,9,21,15,21,27,51;
27, 9,9,21,15,21,27,51,33,21,27,51,51,57,69,117;
81,21,9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279;
(End)

JungHwan Min, Table of n, a(n) for n = 1..5000
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, The movie version
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [_Omar E. Pol_, Nov 02 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A151710, A160171, A160173, A160120, A160715, A161207, A161329, A161331, A161326, A161431, A147582.

YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 1, 3, in == 0 && Count[out, 1] >= 2, 2, in == 0 && Count[out, 1] == 1, 1, True, in]]; A160121[n_] := Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)
A160121[n_] := Count[CellularAutomaton[{13390417258775213635414055181254541831894674613399006361662885886563211940509571858857491972104491013971547937418035084866785430974106432144737472376143620, 4, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)

More terms from David Applegate, Jun 14 2009

A161330 Snowflake (or E-toothpick) sequence (see Comments lines for definition).

Original entry on oeis.org

0, 2, 8, 14, 20, 38, 44, 62, 80, 98, 128, 146, 176, 218, 224, 242, 260, 290, 344, 374, 452, 494, 548, 626, 668, 734, 812, 830, 872, 914, 968, 1058, 1124, 1250, 1340, 1430, 1532, 1598, 1676, 1766, 1856, 1946, 2000, 2066, 2180, 2258, 2384, 2510, 2612, 2714, 2852, 2954, 3116, 3218, 3332, 3494, 3620, 3782, 3896, 3998, 4100

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

This sequence is an E-toothpick sequence (cf. A161328) but starting with two back-to-back E-toothpicks.
On the infinite triangular grid, we start at round 0 with no E-toothpicks.
At round 1 we place two back-to-back E-toothpicks, forming a star with six endpoints.
At round 2 we add six more E-toothpicks.
At round 3 we add six more E-toothpicks.
And so on ... (see the illustrations).
The rule for adding new E-toothpicks is as follows. Each E has three ends, which initially are free. If the ends of two E's meet, those ends are no longer free. To go from round n to round n+1, we add an E-toothpick at each free end (extending that end in the direction it is pointing), subject to the condition that no end of any new E can touch any end of an existing E from round n or earlier. (Two new E's are allowed to touch.)
The sequence gives the number of E-toothpicks in the structure after n rounds. A161331 (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.
Note that, on the infinite triangular grid, a E-toothpick can be represented as a polyedge with three components. In this case, at n-th round, the structure is a polyedge with 3*a(n) components.

David Applegate, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
David Applegate, Illustration of structure after 32 stages. (Contains 1124 E-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.]
Ed Jeffery, Illustration of A161330 structure after 32 stages, with E-toothpicks replace by rhombi (the figure on the right is the complementary structure)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks) [From _Omar E. Pol_, Dec 06 2009]
N. J. A. Sloane, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160120, A160172, A161206, A161328, A161331, A161333.

[No formula or recurrence is known, - N. J. A. Sloane, Oct 13 2023]

For n >= 2, a(n) = 2 + Sum_{k=2..n} 6*A220498(k-1) - 6. - Christopher Hohl, Feb 24 2019. [This is a restatement of the definition. - N. J. A. Sloane, Oct 13 2023]

a(9)-a(12) from N. J. A. Sloane, Dec 07 2012

Corrected and extended by David Applegate, Dec 12 2012

A161207 First differences of A161206.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 12, 12, 18, 24, 30, 30, 28, 30, 20, 12, 18, 26, 34, 42, 50, 56, 54, 44, 48, 64, 82, 80, 68, 66, 36, 12, 18, 26, 34, 42, 50, 58, 58, 54, 66, 90, 114, 126, 122, 120, 102, 60, 48, 70, 94, 118, 142, 160, 162, 136, 130, 160, 204, 198, 160, 142, 68, 12

Offset: 1

Omar E. Pol, Jun 08 2009

nonn

Number of V-toothpicks added to the V-toothpick structure at the n-th round.

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, A160121, A160173, A161206, A161329, A161331.

More terms from R. J. Mathar, Jan 21 2010

A161329 First differences of A161328.

Original entry on oeis.org

1, 3, 5, 7, 13, 11, 17, 15, 21, 23, 25, 27, 33, 27, 25, 15, 25, 35, 41, 55, 53, 59, 61, 59, 65, 63, 57, 47, 37, 47, 65, 71, 97, 95, 105, 95, 89, 83, 81, 87, 93, 79, 73, 79, 89, 107, 113, 119, 113, 115, 117, 135, 125, 127, 129, 135, 153, 135

Offset: 1

Omar E. Pol, Jun 07 2009

nonn

Number of E-Toothpicks added to the E-Toothpick structure at the n-th round.

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. A139251, A160121, A160173, A161328, A161331.

a(8) corrected and more terms added by R. J. Mathar, Jan 21 2010

A161334 Numbers of snowflake sequence, divided by 2: a(n) = A161330(n)/2.

Original entry on oeis.org

0, 1, 4, 7, 10, 19, 22, 31, 40, 49, 64, 73, 88, 109, 112, 121, 130, 145, 172, 187, 226, 247, 274, 313, 334, 367, 406, 415, 436, 457, 484, 529, 562, 625, 670, 715, 766, 799, 838, 883, 928, 973, 1000, 1033, 1090, 1129, 1192, 1255, 1306, 1357, 1426, 1477

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

Also number of E-toothpicks after n-th stage on the semi-infinite triangular grid. A161332 (the first differences) gives the number of E-toothpicks added at n-th stage. - Omar E. Pol, Jan 07 2014

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, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A161328, A161330, A161331, A161332, A161333.

More terms from Omar E. Pol, Jan 07 2014

A211976 First differences of the E-toothpick sequence A211964.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 4, 1, 2, 2, 3, 5, 3, 7, 4, 5, 7, 4, 6, 7, 2, 4, 4, 5, 8, 6, 11, 8, 8, 9, 6, 7, 8, 8, 8, 5, 6, 10, 7, 11, 11, 9, 9, 12, 9, 14, 9, 10, 14, 11, 14, 10, 9, 9, 11, 16, 16, 13, 12, 11, 14, 14, 16, 17, 10, 10, 14

Offset: 1

Omar E. Pol, Dec 15 2012

nonn

Number of E-toothpicks added at n-th stage to the triangular structure of A211964.

Cf. A139250, A139251, A161330, A211964.

a(n) = ((A161331(n+1)/6) + 1)/2.

A220498 Number of E-toothpicks (or tridents) added at n-th stage to the structure of the equilateral triangle of A220478.

Original entry on oeis.org

0, 2, 2, 2, 4, 2, 4, 4, 4, 6, 4, 6, 8, 2, 4, 4, 6, 10, 6, 14, 8, 10, 14, 8, 12, 14, 4, 8, 8, 10, 16, 12, 22, 16, 16, 18, 12, 14, 16, 16, 16, 10, 12, 20, 14, 22, 22, 18, 18, 24, 18, 28, 18, 20, 28, 22, 28, 20, 18, 18, 22, 32, 32, 26, 24, 22, 28, 28, 32, 34, 20, 20, 28

Offset: 0

Omar E. Pol, Feb 19 2013

nonn

Essentially the first differences of A220478.

Cf. A139250, A139251, A160121, A161330, A161329, A161331, A211974, A211976.

a(n) = 1 + A161331(n+1)/6 = 2*A211976(n).

A161337 a(n) = (A161330(n+1) - 2)/3.

Original entry on oeis.org

0, 2, 4, 6, 12, 14, 20, 26, 32, 42, 48, 58, 72, 74, 80, 86, 96, 114, 124, 150, 164, 182, 208, 222, 244, 270, 276, 290, 304, 322, 352, 374, 416, 446, 476, 510, 532, 558, 588, 618, 648, 666, 688, 726, 752, 794, 836, 870, 904, 950, 984, 1038, 1072, 1110, 1164, 1206

Offset: 0

Omar E. Pol, Jun 09 2009

nonn

Cf. A161330, A161331, A161336, A161338.

More terms from Jinyuan Wang, Jul 30 2021

A161338 a(n) = (A161330(n+1) - 2)/2.

Original entry on oeis.org

0, 3, 6, 9, 18, 21, 30, 39, 48, 63, 72, 87, 108, 111, 120, 129, 144, 171, 186, 225, 246, 273, 312, 333, 366, 405, 414, 435, 456, 483, 528, 561, 624, 669, 714, 765, 798, 837, 882, 927, 972, 999, 1032, 1089, 1128, 1191, 1254, 1305, 1356, 1425, 1476, 1557, 1608

Offset: 0

Omar E. Pol, Jun 09 2009

nonn

Cf. A161330, A161331, A161336, A161337.

More terms from Jinyuan Wang, Jul 30 2021

cp's OEIS Frontend

A161332 a(n) = A161331(n)/2.

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A160121 First differences of A160120.

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A161330 Snowflake (or E-toothpick) sequence (see Comments lines for definition).

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A161207 First differences of A161206.

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A161329 First differences of A161328.

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A161334 Numbers of snowflake sequence, divided by 2: a(n) = A161330(n)/2.

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A211976 First differences of the E-toothpick sequence A211964.

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A220498 Number of E-toothpicks (or tridents) added at n-th stage to the structure of the equilateral triangle of A220478.

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A161337 a(n) = (A161330(n+1) - 2)/3.

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A161338 a(n) = (A161330(n+1) - 2)/2.

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