A151710 -id:A151710 - cp's OEIS frontend

A173452 a(n) = A048883(n-1) - A151710(n).

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, -6, 6, 0, 30, 0, 0, 0, 6, -6, 6, 0, 30, -12, 6, 0, 30

Offset: 1

Omar E. Pol, May 29 2010

It appears that the absolute value of a(n) is a multiple of 6, see A008588. - Omar E. Pol, Dec 06 2013

			From _Omar E. Pol_, Dec 06 2013: (Start)
Written as an irregular triangle in which row lengths is A011782 the sequence begins:
0;
0;
0, 0;
0, 0, 0, 6;
0, 0, 0, 6, -6, 6, 0, 30;
0, 0, 0, 6, -6, 6, 0, 30, -12, 6, 0, 30...
(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. [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
Index entries for sequences related to toothpick sequences

Cf. A048883, A139250, A139251, A160121, A151710, A173066, A173067, A173068, A173451, A173453.

A173453 a(n) = A160121(n) - A151710(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 78

Offset: 1

Omar E. Pol, May 29 2010

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. A048883, A139250, A139251, A160121, A151710, A173066, A173067, A173068, A173451, A173452.

A173465 a(n) = A151710(n+1)/3.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 5, 7, 9, 17, 1, 3, 3, 7, 5, 7, 9, 17, 7, 7, 9, 17, 17, 19, 23, 39, 1

Offset: 1

Omar E. Pol, May 29 2010

First 23 terms the same as A151837.

			If written as a triangle, begins:
1;
1,3;
1,3,3,7;
1,3,3,7,5,7,9,17;
1,3,3,7,5,7,9,17,7,7,9,17,17,19,23,39;

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, A151837, A160120, A160121, A160123.

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

A160715 Y-toothpick sequence (as A160120) but without internal propagation in the structure.

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 61, 70, 79, 100, 115, 136, 163, 214, 217, 226, 235, 256, 271, 292, 319, 370, 391, 412, 439, 490, 541, 598, 667, 784, 787

Offset: 0

Omar E. Pol, Jun 02 2009

Note that A160120(9)=67 but in this sequence a(9)=61.

See the entry A160120 for more information about the Y-toothpick sequences.

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. A000079, A000225, A006516, A139250, A160120, A151710.

More terms from Omar E. Pol, May 29 2010

A173451 a(n) = A048883(n-1) - A160121(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 6, -6, 6, 0, 30, -24, 0, 0, 6, -6, 6, 0, 30, -24, 6, 0, 30, -24, 24, 12, 126, -78, -12, 0, 6, -6, 6, 0, 30, -24, 6, 0, 30, -24, 24, 12, 126, -78, -6, 0, 30, -24, 24, 6, 114, -90, 6, 12, 126, -54, 102, 72, 450, -228, -60, 0, 6, -6, 6, 0

Offset: 1

Omar E. Pol, May 29 2010

It appears that the absolute value of a(n) is a multiple of 6, see A008588. - Omar E. Pol, Dec 06 2013

			From _Omar E. Pol_, Dec 06 2013: (Start)
Written as an irregular triangle in which row lengths is A011782 the sequence begins:
.    0;
.    0;
.    0,  0;
.    0,  0,0,6;
.   -6,  0,0,6,-6,6,0,30;
.  -24,  0,0,6,-6,6,0,30,-24,6,0,30,-24,24,12,126;
.  -78,-12,0,6,-6,6,0,30,-24,6,0,30,-24,24,12,126,-78,-6,0,30,-24,24,6,114,-90,6,12,126,-54,102,72,450;
. -228,-60,0,6,-6,6,0...
(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. [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
Index entries for sequences related to toothpick sequences

Cf. A011782, A048883, A139250, A139251, A160120, A160121, A151710, A173066, A173067, A173068, A173452, A173453.

More terms from Nathaniel Johnston, Nov 15 2010

A266533 First differences of A266532.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 3, 9, 9, 21, 9, 21, 21, 45, 3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93, 3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93, 9, 21, 21, 45, 21, 45, 45, 93, 21, 45, 45, 93, 45, 93, 93, 189, 3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93

Offset: 1

David Applegate and Omar E. Pol, Jan 18 2016

Number of Y-toothpicks added at n-th stage in the structure of A266532.

A simplified version of A160121.

			Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
3, 9;
3, 9, 9, 21;
3, 9, 9, 21, 9, 21, 21, 45;
3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93;
...
Observation: at least the first 11 terms of the right border coincide with A068156.

David Applegate, The movie version
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. A000120, A011782, A038573, A068156. A139251, A147582, A160121, A151710, A160721, A266532, A267701.

a(1) = 1. It appears that a(n) = 3*A038573(n-1), n >= 2.

cp's OEIS Frontend

A173452 a(n) = A048883(n-1) - A151710(n).

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A173453 a(n) = A160121(n) - A151710(n).

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A173465 a(n) = A151710(n+1)/3.

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A160121 First differences of A160120.

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A160715 Y-toothpick sequence (as A160120) but without internal propagation in the structure.

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A173451 a(n) = A048883(n-1) - A160121(n).

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A266533 First differences of A266532.

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