A161329 -id:A161329 - cp's OEIS frontend

A160121 First differences of A160120.

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

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

A161328 E-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 16, 29, 40, 57, 72, 93, 116, 141, 168, 201, 228, 253, 268, 293, 328, 369, 424, 477, 536, 597, 656, 721, 784, 841, 888, 925, 972, 1037, 1108, 1205, 1300, 1405, 1500, 1589, 1672, 1753, 1840, 1933, 2012, 2085, 2164, 2253, 2360, 2473, 2592, 2705, 2820

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

An E-toothpick is formed by three toothpicks, as an trident. The E-toothpick has a midpoint and three exposed endpoints such that the distance between the endpoint of the central toothpick and the endpoints of the other toothpicks is equal to 1.
On the infinite triangular grid, we start at round 0 with no E-toothpicks.
At round 1 we place an E-toothpick anywhere in the plane.
At round 2 we add three more E-toothpicks.
At round 3 we add five more E-toothpicks.
And so on... (see 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. A161329 (the first differences) gives the number added at the n-th round.
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. See the entry A139250 for more information about the growth of the toothpicks.
See also the snowflake sequence A161330.

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.]
Dr. Goulu, 2012, Mayas, Kaya et cure-dents, Pourquoi Comment Combien blog, January 2012 (in French).
Omar E. Pol, A magic wand with star in the E-toothpick cellular automaton
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330
N. J. A. Sloane, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Zozoped, Illustration of the structure, a(42) = 2012 [Broken link].
Zozoped, Illustration of the structure, a(42) = 2012, "Nous avons vu se lever son étoile", Le blog du Barabel [broken link].
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160120, A160172, A161206, A161329, A161330.

For n >= 3, a(n) = 4 + Sum_{k=3..n} 2*Sum_{x=1..3} A220498(k-x) + 2^((k mod 2) + 1) - 7. - Christopher Hohl, Feb 24 2019

a(8) corrected, more terms appended by R. J. Mathar, Jan 21 2010
Extensive edits by Omar E. Pol, May 14 2012
I have copied the rule for adding new E-toothpicks (described by N. J. A. Sloane) from A161330. - Omar E. Pol, Dec 07 2012

A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

Original entry on oeis.org

0, 1, 3, 5, 9, 9, 9, 13, 25, 21, 9, 13, 25, 25, 25, 37, 73, 57, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 165, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 169, 25, 37, 73, 73, 73, 109, 217, 181, 73, 109, 217, 217, 217, 325, 649, 489, 9, 13, 25

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

Essentially the first differences of A160172.
For further information see the Applegate-Pol-Sloane paper, chapter 11: T-shaped toothpicks. See also the figure 16 in the mentioned paper. - Omar E. Pol, Nov 18 2011
The numbers n in increasing order such that the triple [n, n, n] can be found here, give A199111. [Observed by Omar E. Pol, Nov 18 2011. Confirmed by Alois P. Heinz, Nov 21 2011]

			From _Omar E. Pol_, Feb 09 2010: (Start)
If written as a triangle:
0;
1;
3;
5;
9,9;
9,13,25,21;
9,13,25,25,25,37,73,57;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,165;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109,217,181,73,109,217,217,217,325,649,489;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109...
(End)

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

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.], which is also available at arXiv:1004.3036v2
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, A160172, A160121, A160171, A161207, A161329, A172311.

wt[n_] := DigitCount[n, 2, 1];
a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := 2/3 (3^wt[n-1] + 3^wt[n-2]) + 1;
Table[a[n], {n, 0, 68}] (* Jean-François Alcover, Aug 18 2018, after N. J. A. Sloane *)

a(n) = (2/3)*(3^wt(n-1) + 3^wt(n-2))+1 (where wt is A000120), for n >= 3. - N. J. A. Sloane, Jan 01 2010

More terms from N. J. A. Sloane, Jan 01 2010

A172311 First differences of A172310.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 14, 18, 18, 20, 24, 24, 38, 34, 42, 34, 26, 28, 32, 38, 52, 54, 64, 58, 68, 60, 60, 50, 66, 70, 70, 74, 50, 52, 60, 54, 64, 66, 84, 88, 116, 106, 132, 100, 136, 126, 140, 106, 118, 100, 122, 106, 138, 114, 138, 132, 152, 156, 176, 158, 190, 166, 158, 154, 98, 88, 132, 82, 124, 94, 112

Offset: 0

Omar E. Pol, Jan 31 2010

nonn

Number of L-toothpicks added to the L-toothpick structure of A172310 at the n-th stage.

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, A160171, A160173, A161207, A161329, A172310.

More terms from Nathaniel Johnston, Nov 15 2010

Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

A161331 First differences of A161330.

Original entry on oeis.org

0, 2, 6, 6, 6, 18, 6, 18, 18, 18, 30, 18, 30, 42, 6, 18, 18, 30, 54, 30, 78, 42, 54, 78, 42, 66, 78, 18, 42, 42, 54, 90, 66, 126, 90, 90, 102, 66, 78, 90, 90, 90, 54, 66, 114, 78, 126, 126, 102, 102, 138, 102, 162, 102, 114, 162, 126, 162, 114, 102, 102, 126, 186, 186, 150, 138, 126, 162, 162, 186, 198, 114, 114, 162

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

Number of E-toothpicks added to the snowflake structure at n-th round.

David Applegate, Table of n, a(n) for n = 0..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, A160121, A160173, A161328, A161329, A161330.

More terms from David Applegate, Dec 13 2012

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).

A161413 First differences of A161412.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 3, 4, 6, 4, 5, 5, 10, 9, 5

Offset: 1

Omar E. Pol, Jun 10 2009

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

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, A160407, A161207, A161329, A161412.

A294961 a(n) is the number of E-toothpicks added at n-th stage in the structure of the snowflake cellular automaton of A294960.

Original entry on oeis.org

2, 6, 6, 6, 6, 18, 6, 18, 18, 18, 6, 18, 30, 18, 30, 54, 18, 42, 30, 42, 18, 42, 42, 54, 66

Offset: 1

Omar E. Pol, Feb 10 2018

First differences of A294960.

Cf. A139251, A161328, A161329, A161330, A161331.

cp's OEIS Frontend

A160121 First differences of A160120.

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A161207 First differences of A161206.

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A161328 E-toothpick sequence (see Comments lines for definition).

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A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

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A172311 First differences of A172310.

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A161331 First differences of A161330.

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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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A161413 First differences of A161412.

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A294961 a(n) is the number of E-toothpicks added at n-th stage in the structure of the snowflake cellular automaton of A294960.

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