A194441 -id:A194441 - cp's OEIS frontend

A194435 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194434.

0, 4, 8, 16, 16, 16, 32, 44, 32, 16, 32, 64, 96, 48, 80, 100, 64, 16, 32, 64, 96, 112, 144, 168, 176, 80, 96, 160, 256, 128, 176, 212, 128, 16, 32, 64, 96, 112, 144, 176, 208, 168, 192, 240, 400, 272, 336, 332, 336, 112, 96, 176, 288, 336, 416, 464

Offset: 0

Omar E. Pol, Sep 03 2011

Essentially the first differences of A194434.
First differs from A221528 at a(13). - Omar E. Pol, Mar 23 2013
From Omar E. Pol, Jun 24 2022: (Start)
The word of this cellular automaton is "ab".
For the nonzero terms the structure of the irregular triangle is as shown below:
  a,b;
  a,b;
  a,b,a,b;
  a,b,a,b,a,b,a,b;
  a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
  ...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of D-toothpicks (of length sqrt(2)).
Columns "b" contain numbers of toothpicks (of length 1).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
For further information about the word of cellular automata see A296612. (End)

			From _Omar E. Pol_, Mar 23 2013: (Start)
When written as an irregular triangle the sequence of nonzeros terms begins:
   4, 8;
  16,16;
  16,32,44,32;
  16,32,64,96, 48, 80,100, 64;
  16,32,64,96,112,144,168,176, 80, 96,160,256,128,176,212,128;
  16,32,64,96,112,144,176,208,168,192,240,400,272,336,332,336,112,96, ...
  ... (End)
Right border gives the powers of 2 >= 8 (reformatted the triangle). - _Omar E. Pol_, Jun 24 2022

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animated version for n = 0..31 (red elements)
Paolo Xausa, Animated version for n = 0..63 (red elements)
Paolo Xausa, Illustration of initial terms for n = 0..63 (red elements, multipage PDF)
Index entries for sequences related to toothpick sequences

Cf. A011782, A139251, A194271, A194433, A194434, A194441, A194443, A194445, A212009, A221528, A299612.

a(n) = 4*A194445(n).

Conjecture: a(2^k+1) = 16, if k >= 1.

More terms from Omar E. Pol, Mar 23 2013

A194693 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194692.

Original entry on oeis.org

2, 2, 4, 6, 8, 8, 12, 13, 12, 10, 16, 20, 32, 20, 24, 27, 20, 10, 16, 24, 36, 41, 48, 48, 54, 34, 44, 50, 80, 48, 48, 55, 36, 10, 16, 24, 36, 41, 48, 54, 68, 62, 80, 80, 124, 95, 104, 89, 94, 44, 48, 68

Offset: 1

Omar E. Pol, Sep 03 2011

Sequence related to the D-toothpick "wide" triangle (See A194440 and A194441). First differences of A194692.

			Written as a triangle:
2,
2,
4,
6,8,
8,12,13,12,
10,16,20,32,20,24,27,20,
10,16,24,36,41,48,48,54,34,44,50,80,48,48,55,36,
10,16,24,36,41,48,54,68,62,80,80,124,95,104,89,94,44,...

Cf. A086224, A152980, A194271, A194440, A194441, A194692, A194695, A194696

a(n) = A194696(n)/2.
Conjectures for n = 2^k+j, if 0<=j<=3:
a(2^k) = 7*2^(k-2)-1, if k >= 2.
a(2^k+1) = 2^k+4, if k >= 2.
a(2^k+2) = 10, if k >= 3.
a(2^k+3) = 16, if k >= 3.
End conjectures.

A194696 a(n) = 2*A194693(n).

Original entry on oeis.org

4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 20, 32, 48, 72, 82, 96, 96, 108, 68, 88, 100, 160, 96, 96, 110, 72, 20, 32, 48, 72, 82, 96, 108, 136, 124, 160, 160, 248, 190, 208, 178, 188, 88, 96, 136

Offset: 1

Omar E. Pol, Sep 03 2011

Conjecture: number of toothpicks or D-toothpicks added to the structure of A194440 at stage 2^k+n, if k tends to infinity. It appears that rows of A194441 when written as a triangle converge to this sequence.

			Written as a triangle:
4,
4,
8,
12,16,
16,24,26,24,
20,32,40,64,40,48,54,40,
20,32,48,72,82,96,96,108,68,88,100,160,96,96,110,72,
20,32,48,72,82,96,108,136,124,160,160,248,190,208,178,...

Cf. A152980, A194270, A194440, A194692, A194693, A194697.

A220527 Number of toothpicks or D-toothpicks added at n-th stage in the toothpick structure of A220526.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 14, 12, 15, 16, 4, 4, 8, 12, 16, 10, 24, 26

Offset: 0

Omar E. Pol, Jan 02 2013

Essentially the first differences of A220526.

			Written as an irregular triangle begins:
0;
1;
2;
4,4;
4,4,7,8;
4,4,8,12,14,12,15,16;
4,4,8,12,16,10,24,26,...

Cf. A139250, A139251, A194271, A194441, A194443, A220501, A220521, A220523, A220526.

A212009 Number of toothpicks or D-toothpicks added at n-th stage in the toothpick structure of A212008.

Original entry on oeis.org

0, 1, 4, 8, 16, 22, 20, 24, 36, 40, 32, 44, 56, 94, 60, 56, 76, 72, 32, 48, 72, 112, 132, 140, 136, 168, 112, 132, 140, 238, 148, 120, 156, 136, 32, 48, 72, 112, 132

Offset: 0

Omar E. Pol, Dec 15 2012

Essentially the first differences of A212008.

			When written as an irregular triangle:
0;
1;
4;
8;
16,22;
20,24,36,40;
32,44,56,94,60,56,76,72;
32,48,72,112,132,140,136,168,112,132,140,238,148,120,156,136;
32,48,72,112,132,...

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 toothpick sequences

Cf. A139250, A139251, A194271, A194433, A194435, A194441, A194443, A194445, A220501.

It appears that a(2^k + 2) = 32, if k >= 3.

A220495 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220494.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32, 32, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32, 32, 8, 8, 16, 16, 16, 16, 32, 32, 16, 16, 32, 32, 32, 32, 64, 64, 4, 4, 8, 8, 8, 8, 16, 16

Offset: 0

Omar E. Pol, Dec 23 2012

Essentially the first differences of A220494.

First differs from A194441 (and from A220521) at a(12).

			When written as an irregular triangle begins:
0;
1;
2;
4,4;
4,4,8,8;
4,4,8,8,8,8,16,16;
4,4,8,8,8,8,16,16,8,8,16,16,16,16,32,32;
4,4,8,8,8,8,16,16,8,8,16,16,16,16,32,32,8,8,16,16,...

Cf. A139250, A139251, A194271, A194701, A194441, A220494, A220497, A220521.

A233777 Number of vertices in the n-th row of the toothpick structure of A233776, with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 4, 5, 2, 4, 4, 6, 4, 8, 8, 9, 2, 4, 4, 6, 4, 8, 8, 10, 4, 8, 8, 12, 8, 16, 16, 17, 2, 4, 4, 6, 4, 8, 8, 10, 4, 8, 8, 12, 8, 16, 16, 18, 4, 8, 8, 12, 8, 16, 16, 20, 8, 16, 16, 24, 16, 32, 32, 33, 2, 4, 4, 6, 4, 8, 8, 10, 4, 8, 8, 12

Offset: 0

Omar E. Pol, Dec 25 2013

			Illustration of initial terms:
---------------------------------------------------------
. A001316 A006046         Diagram          a(n)  A233776
---------------------------------------------------------
.            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
.
Written as an irregular triangle the sequence begins:
1;
1;
2;
2,3;
2,4,4,5;
2,4,4,6,4,8,8,9;
2,4,4,6,4,8,8,10,4,8,8,12,8,16,16,17;
2,4,4,6,4,8,8,10,4,8,8,12,8,16,16,18,4,8,8,12,8,16,16,20,8,16,16,24,16,32,32,33;

Cf. A001316, A139251, A194271, A194441, A233775, A233776.

cp's OEIS Frontend

A194435 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194434.

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A194693 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194692.

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A194696 a(n) = 2*A194693(n).

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A220527 Number of toothpicks or D-toothpicks added at n-th stage in the toothpick structure of A220526.

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A212009 Number of toothpicks or D-toothpicks added at n-th stage in the toothpick structure of A212008.

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A220495 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220494.

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A233777 Number of vertices in the n-th row of the toothpick structure of A233776, with a(0) = 1.

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