A162796 -id:A162796 - cp's OEIS frontend

A163094 a(n) = A162796(n)/2.

0, 1, 3, 7, 11, 15, 21, 35, 43, 47, 53, 67, 77, 91, 111, 155, 171, 175, 181, 195, 205, 219, 239, 283, 301, 315, 335, 379, 407, 453, 523, 651, 683, 687, 693, 707, 717, 731, 751, 795, 813, 827, 847, 891, 919, 965, 1035, 1163, 1197, 1211, 1231, 1275, 1303, 1349, 1419

Offset: 0

Omar E. Pol, Jul 28 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, A139251, A159791, A159792, A162793, A162794, A162795, A162796, A162797.

a(0) prepended by and more terms from Jinyuan Wang, Mar 15 2020

A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 53, 85, 89, 101, 117, 149, 165, 201, 261, 341, 345, 357, 373, 405, 421, 457, 517, 597, 613, 649, 709, 793, 853, 965, 1173, 1365, 1369, 1381, 1397, 1429, 1445, 1481, 1541, 1621, 1637, 1673, 1733, 1817, 1877, 1989, 2197, 2389, 2405, 2441, 2501

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

Partial sums of A162793.
Also, total number of ON cells at stage n of the two-dimensional cellular automaton defined as follows: replace every "vertical" toothpick of length 2 with a centered unit square "ON" cell, so we have a cellular automaton which is similar to both A147562 and A169707 (this is the "one-step bishop" version). For the "one-step rook" version we use toothpicks of length sqrt(2), then rotate the structure 45 degrees and then replace every toothpick with a unit square "ON" cell. For the illustration of the sequence as a cellular automaton we now have three versions: the original version with toothpicks, the one-step rook version and one-step bishop version. Note that the last two versions refer to the standard ON cells in the same way as the two versions of A147562 and the two versions of A169707. It appears that the graph of this sequence lies between the graphs of A147562 and A169707. Also, it appears that this sequence shares infinitely many terms with both A147562 and A169707, see Formula section and Example section. - Omar E. Pol, Feb 20 2015
It appears that this is also a bisection (the odd terms) of A255747.

			From _Omar E. Pol_, Feb 18 2015: (Start)
Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
    1;
    5;
    9, 21;
   25, 37, 53, 85;
   89,101,117,149,165,201,261,341;
  345,357,373,405,421,457,517,597,613,649,709,793,853,965,1173,1365;
  ...
The right border gives the positive terms of A002450.
(End)
It appears that T(j,k) = A147562(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements of the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015

Michel Marcus, Table of n, a(n) for n = 1..10000
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. A002450, A048645, A139250, A139251, A147562, A153000, A159791, A159792, A160164, A160552, A162793, A162794, A162796, A162797, A169707, A255263, A255264, A255747.

It appears that a(n) = A147562(n) = A169707(n), if n is a term of A048645, otherwise A147562(n) < a(n) < A169707(n). - Omar E. Pol, Feb 20 2015
It appears that a(n) = (A169707(2n) - 1)/4 = A255747(2n-1). - Omar E. Pol, Mar 07 2015
a(n) = 1 + 4*A255737(n-1). - Omar E. Pol, Mar 08 2015

More terms from N. J. A. Sloane, Dec 28 2009

A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 17, 1, 5, 5, 17, 5, 17, 21, 49, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 5, 17, 21, 49, 21, 53, 81, 129

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

It appears that a(2^k) = 1, for k >= 0. [From Omar E. Pol, Feb 22 2010]

			Contribution from _Omar E. Pol_, Feb 22 2010: (Start)
If written as a triangle:
1;
1,5;
1,5,5,17;
1,5,5,17,5,17,21,49;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129,5,17,21...
Rows converge to A173464.
(End)
Contribution from Omar E. Pol, Apr 01 2011 (Start):
It appears that the final terms of rows give A000337.
It appears that row sums give A006516.
(End)

Nathaniel Johnston, Table of n, a(n) for n = 1..94
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, A159791, A159792, A162793, A162794, A162795, A162796.

Cf. A000337, A058922, A173464. [From Omar E. Pol, Feb 22 2010]

a(n) = A162796(n) - A162795(n).

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Feb 22 2010
More terms (a(51)-a(55)) from Nathaniel Johnston, Mar 30 2011

A162793 Number of toothpicks added to the toothpick structure A139250 at the n-th odd round.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 16, 32, 4, 12, 16, 32, 16, 36, 60, 80, 4, 12, 16, 32, 16, 36, 60, 80, 16, 36, 60, 84, 60, 112, 208, 192, 4, 12, 16, 32, 16, 36, 60, 80, 16, 36, 60, 84, 60, 112, 208, 192, 16, 36, 60, 84, 60, 112, 208, 196, 60, 112, 208, 224, 212, 364, 672, 448, 4, 12, 16, 32, 16

Offset: 1

Omar E. Pol, Jul 14 2009

Bisection of A139251.
Note that these toothpicks are parallel to the initial toothpick in the structure.
First differences of A162795. - Omar E. Pol, Feb 23 2015

			From _Omar E. Pol_, Feb 23 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,12;
4,12,16,32;
4,12,16,32,16,36,60,80;
4,12,16,32,16,36,60,80,16,36,60,84,60,112,208,192;
4,12,16,32,16,36,60,80,16,36,60,84,60,112,208,192,16,36,60,84,60,112,208,196,60,112,208,224,212,364,672,448;
...
It appears that right border gives the positive terms of A001787.
It appears that row sums give A000302.
(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 cellular automata

Cf. A000302, A001787, A139250, A139251, A147582, A159791, A159792, A162794, A162795, A162796, A162797, A169708.

More terms from N. J. A. Sloane, Dec 28 2009

A162794 Number of toothpicks added to the toothpick structure A139250 at the n-th even round.

Original entry on oeis.org

0, 2, 4, 8, 8, 8, 12, 28, 16, 8, 12, 28, 20, 28, 40, 88, 32, 8, 12, 28, 20, 28, 40, 88, 36, 28, 40, 88, 56, 92, 140, 256, 64, 8, 12, 28, 20, 28, 40, 88, 36, 28, 40, 88, 56, 92, 140, 256, 68, 28, 40, 88, 56, 92, 140, 256, 88, 92, 140, 260, 172, 296, 488, 704, 128, 8, 12, 28, 20, 28

Offset: 0

Omar E. Pol, Jul 14 2009

nonn

Note that these toothpicks are orthogonal to the initial toothpick in the sieve.

A bisection of A139251.

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, A159791, A159792, A162793, A162795, A162796, A162797.

Extended by R. J. Mathar, Sep 27 2009

A255263 Differences between the total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood and the total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 12, 20, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 4, 12, 20, 12, 36, 80, 68, 12, 36, 80, 84, 96, 208, 352, 196, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 4, 12, 20, 12, 36, 80, 68, 12, 36, 80

Offset: 1

Omar E. Pol, Feb 19 2015

It appears that the graph of A162795 lies between the graphs of A147562 and A169707.

It appears that a(n) = 0 if and only if n is a member of A048645.

			Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
0;
0;
0,0;
0,0,4,0;
0,0,4,0,4,12,20,0;
0,0,4,0,4,12,20,0,4,12,20,12,36,80,68,0;
0,0,4,0,4,12,20,0,4,12,20,12,36,80,68,0,4,12,20,12,36,80,68,12,36,80,84,96,208,352,196,0;
...
It appears that if k is a power of 2 then T(j,k) = 0.

Cf. A011782, A048645, A139250, A160164, A162796, A169707, A147562, A255166, A255264.

a(n) = A169707(n) - A162795(n).

A194800 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 2 and the horizontal toothpicks have length 4.

Original entry on oeis.org

0, 3, 11, 17, 31, 39, 67

Offset: 0

Omar E. Pol, Sep 07 2011

There are an infinite family of these sequences since A139250 gives the number of toothpicks in the structure regardless of the length difference between horizontal toothpicks and vertical toothpicks. Examples: A147614, this sequence, A194802, A160420, etc.

			a(2) = 11.
o o o o o
. . o . .
o o o o o

Cf. A139250, A147614, A162795, A162796, A162797, A175098, A160420, A160422, A194802.

A194802 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 4 and the horizontal toothpicks have length 2.

Original entry on oeis.org

0, 5, 9, 23, 29, 45, 57

Offset: 0

Omar E. Pol, Sep 07 2011

There are an infinite family of these sequences since A139250 gives the number of toothpicks in the structure regardless of the length difference between horizontal toothpicks and vertical toothpicks. Examples: A147614, A194800, this sequence, A160420, etc.

			a(2) = 9.
o o o
. o .
. o .
. o .
o o o

Cf. A139250, A147614, A162795, A162796, A162797, A175098, A160420, A160422, A194800

A173464 When regarded as a triangle, the rows of A162797 converge to this sequence.

Original entry on oeis.org

1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 5, 17, 21, 49, 21, 53, 81, 129, 21, 53, 81, 133, 81, 165, 289, 321

Offset: 1

Omar E. Pol, Feb 22 2010

			From _Omar E. Pol_, Apr 01 2011: (Start)
If written as a triangle begins:
1,
5,
5,17,
5,17,21,49,
5,17,21,49,21,53,81,129,
5,17,21,49,21,53,81,129,21,53,81,133,81,165,289,321,
...
It appears that the final terms of rows give A000337.
It appears that row sums give A010036.
(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

Cf. A139250, A162795, A162796, A162797.

Cf. A010036, A162793, A162794. - Omar E. Pol, Apr 01 2011

More terms a(20)-a(32) from Omar E. Pol, Apr 01 2011

cp's OEIS Frontend

A163094 a(n) = A162796(n)/2.

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A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

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A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

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A162793 Number of toothpicks added to the toothpick structure A139250 at the n-th odd round.

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A162794 Number of toothpicks added to the toothpick structure A139250 at the n-th even round.

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A255263 Differences between the total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood and the total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

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Formula

A194800 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 2 and the horizontal toothpicks have length 4.

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A194802 Number of grid points that are covered after n-th stage of A139250, assuming the vertical toothpicks have length 4 and the horizontal toothpicks have length 2.

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A173464 When regarded as a triangle, the rows of A162797 converge to this sequence.

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