A160128 -id:A160128 - cp's OEIS frontend

A139252 Number of segments needed to draw the toothpick structure of A139250 as it is after n stages.

0, 1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 31, 39, 51, 59, 61, 63, 67, 71, 75, 83, 95, 103, 107, 115, 127, 139, 155, 183, 215, 231, 233, 235, 239, 243, 247, 255, 267, 275, 279, 287, 299, 311, 327, 355, 387, 403, 407, 415

Offset: 0

Omar E. Pol, May 17 2008

nonn

Contribution from Omar E. Pol, Sep 16 2012 (Start):
It appears that A147614(n)/a(n) converge to 4.
It appears that A139250(n)/a(n) converge to 3.
It appears that A160124(n)/a(n) converge to 2.
(End)

			For n = 3, after three stages the toothpick structure of A139250 contains seven toothpicks (A139250(3) = 7), however the toothpick structure can be essentially represented by five segments, so a(3) = 5. - _Omar E. Pol_, Sep 16 2012

Nathaniel Johnston, Table of n, a(n) for n = 0..256
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.]
Nathaniel Johnston, C script for computing table of terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A139253, A147614, A160124, A139254, A139255, A160128.

Terms after a(28) from Nathaniel Johnston, Mar 29 2011

A160425 a(n) = number of grid points that are covered after n-th rounds of A160120.

Original entry on oeis.org

0, 4, 10, 19, 31, 40

Offset: 0

Omar E. Pol, Jun 01 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, A147614, A160120, A160121, A160128.

A194811 Number of grid points that are covered after n-th stage of A139250 version "Tree", starting with a(0) = 1 and assuming the toothpicks have length 4, 3, and 2.

Original entry on oeis.org

1, 5, 13, 25, 37, 53, 81, 113, 133, 149, 177, 213, 253, 313, 401, 481, 517, 533, 561, 597, 637, 697, 785, 869, 925, 985, 1077, 1189, 1329, 1537, 1793, 1985, 2053, 2069, 2097, 2133, 2173, 2233, 2321, 2405, 2461, 2521, 2613, 2725, 2865, 3073, 3329, 3525, 3613

Offset: 0

Omar E. Pol, Oct 24 2011

nonn

The first differences give A147646.

Cf. A139250, A152980, A153006, A147646, A159795, A160128, A160420, A160422, A160425, A182618, A194800, A194802.

a(n) = 1 + 4*A153006(n) = 1 + A159795(n).

A210985 Number of segments needed to draw the toothpick structure of A139250 as it is after 2^n stages.

Original entry on oeis.org

1, 3, 7, 19, 63, 235, 919, 3651, 14575

Offset: 0

Omar E. Pol, Sep 11 2012

It appears that this is also the partial sums of A039301.

It appears that this is also 1 together with A160128.

			For n = 3, after 2^3 stages the toothpick structure of A139250 contains 43 toothpicks (A139250(2^3) = 43). However the toothpick structure can be essentially represented by 19 segments (A139252(2^3) = 19), so a(3) = 19.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A007583, A039301, A139250, A139252, A147614, A160128.

a(n) = A139252(2^n).

A233776 Number of grid points that are covered on the semi-infinite hexagonal grid after n-th stage in a cellular automaton in which the toothpicks are connected by their endpoints but the toothpicks placed in northwest or northeast direction are prohibited, starting with a(0) = 1.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 15, 19, 24, 26, 30, 34, 40, 44, 52, 60, 69, 71, 75, 79, 85, 89, 97, 105, 115, 119, 127, 135, 147, 155, 171, 187, 204, 206, 210, 214, 220, 224, 232, 240, 250, 254, 262, 270, 282, 290, 306, 322, 340, 344, 352, 360, 372, 380, 396, 412, 432, 440, 456, 472, 496

Offset: 0

Omar E. Pol, Dec 25 2013

nonn

a(n) is also the number of grid points that are covered on the semi-infinite square grid after n-th stage in a D-toothpick structure in which the toothpicks placed in horizontal direction are prohibited, starting with a(0) = 1.

Note that both the structure mentioned in the above comment and the structure mentioned in the definition are essentially equivalents. Both structures are related to the Sierpinski triangle (see A047999). The number of edges in the structure after n-th stage is A006046(n). The number of edges added at n-th stage is A001316(n-1), for n >= 1 (see example).

			Illustration of initial terms:
---------------------------------------------------------
. A001316 A006046         Diagram         A233777   a(n)
---------------------------------------------------------
.            0
.                            *               1       1
.    1       1               |
.                            *               1       2
.    2       3              / \
.                          *   *             2       4
.    2       5             |   |
.                          *   *             2       6
.    4       9            / \ / \
.                        *   *   *           3       9
.    2      11           |       |
.                        *       *           2      11
.    4      15          / \     / \
.                      *   *   *   *         4      15
.    4      19         |   |   |   |
.                      *   *   *   *         4      19
.    8      27        / \ / \ / \ / \
.                    *   *   *   *   *       5      24
.    2      29       |               |
.                    *               *       2      26
.    4      33      / \             / \
.                  *   *           *   *     4      30
.

Omar E. Pol, Illustration of the structure on the square grid after 32 stages
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Partial sums of A233777.

Cf. A001316, A006046, A047999, A139250, A141614, A160120, A160128, A160425, A194270, A194440, A233774.

cp's OEIS Frontend

A139252 Number of segments needed to draw the toothpick structure of A139250 as it is after n stages.

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A160425 a(n) = number of grid points that are covered after n-th rounds of A160120.

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A194811 Number of grid points that are covered after n-th stage of A139250 version "Tree", starting with a(0) = 1 and assuming the toothpicks have length 4, 3, and 2.

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A210985 Number of segments needed to draw the toothpick structure of A139250 as it is after 2^n stages.

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A233776 Number of grid points that are covered on the semi-infinite hexagonal grid after n-th stage in a cellular automaton in which the toothpicks are connected by their endpoints but the toothpicks placed in northwest or northeast direction are prohibited, starting with a(0) = 1.

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