A194445 -id:A194445 - cp's OEIS frontend

A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

0, 1, 4, 8, 16, 22, 24, 22, 40, 40, 32, 32, 56, 74, 96, 50, 88, 72, 32, 48, 72, 104, 128, 112, 144, 144, 152, 96, 152, 178, 240, 122, 184, 136, 32, 48, 72, 108, 144, 144, 184, 188, 200, 176, 272, 274, 416, 250, 288, 272, 216, 144, 208, 292, 384, 332, 376

Offset: 0

Omar E. Pol, Aug 23 2011

nonn

Essentially the first differences of A194270.

			Written as a triangle:
0,
1,
4,
8,
16,22,
24,22,40,40,
32,32,56,74,96,50,88,72,
32,48,72,104,128,112,144,144,152,96,152,178,240,122,184,136,
32,48,72,108,144,144,184,188,200,176,272,274,416,250,288,...

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. A010694, A129370, A139250, A139251, A172311, A182839, A194270, A194441, A194443, A194445.

a(n) = n^2-(n-1)^2*(1-(-1)^n)/8, if 0 <= n <=4.
Let b(n) = A194441(n), let c(n) = A194443(n), let d(n) = A010694(n), then:
Conjecture: a(n) = 4*(b(n-1)-d(n)) + 2*(c(n)-d(n+1)) + 2*(c(n+2)-d(n+1)) + 8, if n >= 3.
Conjecture: a(2^k+2) = 32, if k >= 3.

More terms from Omar E. Pol, Sep 01 2011

A194441 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194440.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 12, 16, 8, 16, 16, 4, 4, 8, 12, 16, 16, 24, 26, 24, 12, 20, 28, 40, 20, 32, 32, 4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 12, 20, 32, 48, 48, 64, 70, 76, 30, 44, 64, 88, 44, 64, 64, 4, 4, 8, 12

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

Essentially the first differences of A194440.

			If written as a triangle:
0,
1,
2,
4,4,
4,4,8,8,
4,4,8,12,16,8,16,16,
4,4,8,12,16,16,24,26,24,12,20,28,40,20,32,32,
4,4,8,12,16,16,24,26,24,20,32,40,64,40,48,54,40,12,20,...
.
It appears that rows converge to A194696.

Cf. A139251, A160121, A160407, A161831, A194271, A194440, A194443, A194445, A194692, A194693, A194696.

Conjectures for n = 2^k+j, if -1<=j<=3:
a(2^k-1) = 2^k, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 2.
End of conjectures.

More terms from Omar E. Pol, Dec 28 2012

A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 8, 8, 13, 16, 4, 4, 8, 12, 16, 16, 20, 24, 12, 8, 16, 28, 16, 16, 25, 32, 4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 32, 40, 33, 48, 20, 8, 16, 28, 40, 44, 50, 60, 28, 16, 32, 60, 32, 32, 49, 64, 4, 4, 8

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

Essentially the first differences of A194442. It appears that the structure of the "narrow" triangle is much more regular about n=2^k, see formula section.

			If written as a triangle:
0,
1,
2,
4,4,
4,4,7,8,
4,4,8,12,8,8,13,16,
4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32,
4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28...
.
It appears that rows converge to A194697.

Cf. A139251, A160121, A160407, A161831, A194271, A194441, A194442, A194445, A194694, A194695, A194697.

Conjectures for n = 2^k+j, if -6<=j<=6:
a(2^k-6) = 2^(k-2), if k >= 3.
a(2^k-5) = 2^(k-1), if k >= 3.
a(2^k-4) = 2^k-4, if k >= 2.
a(2^k-3) = 2^(k-1), if k >= 3.
a(2^k-2) = 2^(k-1), if k >= 2.
a(2^k-1) = 3*2^(k-2)+1, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 3.
a(2^k+4) = 12, if k >= 3.
a(2^k+5) = 16, if k >= 4.
a(2^k+6) = 16, if k >= 4.
End of conjectures.

A194444 D-toothpick sequence of the second kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 106, 126, 151, 167, 171, 179, 195, 219, 247, 283, 325, 369, 389, 413, 453, 517, 549, 593, 646, 678, 682, 690, 706, 730, 758, 794, 838, 890, 932, 980, 1040, 1140, 1208, 1292, 1375, 1459, 1487, 1511, 1555

Offset: 0

Omar E. Pol, Aug 24 2011

nonn

This cellular automaton has essentially the same rules as A194270. We start at stage 0 with no toothpicks. At stage 1, we place a D-toothpick of length sqrt(2), in diagonal direction, at (0,0),(1,1). At stage 2, we place two toothpicks of length 1. At stage 3 we place four D-toothpicks. And so on. The toothpicks and D-toothpicks are connected by their endpoints. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194445) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure shows a fractal (or fractal-like) behavior.

First differs from A220524 at a(13). - Omar E. Pol, Mar 23 2013

Cf. A139250, A172310, A182838, A194270, A194432, A194434, A194440, A194442, A194445, A212008, A220524.

a(n) = A194434(n)/4. - Omar E. Pol, Oct 15 2011

More terms from Omar E. Pol, Mar 23 2013

A182635 Number of toothpicks added at n-th stage to the toothpick structure of A182634.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 16, 20, 16, 20, 28, 16, 4, 8, 16, 20, 20, 28, 44, 44, 24, 20, 40, 56, 48, 52, 64, 32, 4, 8, 16, 20, 20, 28, 44, 44, 28, 28, 52, 76, 76, 76

Offset: 0

Omar E. Pol, Dec 08 2010

First differences of A182634.

First differs from A139251 at a(11).

			Contribution from _Omar E. Pol_, Dec 06 2012 (Start):
When written as an irregular triangle begins:
0;
1;
2;
4,4;
4,8,12,8;
4,8,16,20,16,20,28,16;
4,8,16,20,20,28,44,44,24,20,40,56,48,52,64,32;
4,8,16,20,20,28,44,44,28,28,52,76,76,76,...
(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
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160121, A161207, A161645, A182633, A182634, A194445.

a(n) = A182633(n)/3.

A220523 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "narrow" triangle of A220522.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 8, 8, 13, 16, 4, 4, 8, 12, 16, 16, 20, 24, 12, 8, 16, 28, 16, 16, 25, 32, 4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 32, 40, 41, 48, 20, 8

Offset: 0

Omar E. Pol, Dec 15 2012

Essentially the first differences of A220522. First differs from A194443 at a(47).

			Written as an irregular triangle begins:
0;
1;
2;
4,4;
4,4,7,8;
4,4,8,12,8,8,13,16;
4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32;
4,4,8,12,16,16,22,32,26,20,24,40,32,40,41,48,20,8,...

Cf. A139250, A139251, A194441, A194443, A194445, A220521, A220522.

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

Original entry on oeis.org

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

A220525 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220524.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 11, 8, 4, 8, 16, 24, 16, 20, 25, 16, 4, 8, 16, 24, 32, 36, 42, 44, 20, 24, 44, 64, 40, 44, 53, 32, 4, 8, 16, 24, 32, 36, 44, 52, 42, 48, 64, 100, 80, 84, 91, 84, 28, 24, 48, 72, 92, 104, 124, 132, 54, 56, 100, 144, 88, 92, 109, 64, 4

Offset: 0

Omar E. Pol, Mar 23 2013

Essentially the first differences of A220524.

First differs from A194445 at a(13).

			When written as an irregular triangle begins:
0;
1;
2;
4,4;
4,8,11,8;
4,8,16,24,16,20,25,16;
4,8,16,24,32,36,42,44,20,24,44,64,40,44,53,32;
4,8,16,24,32,36,44,52,42,48,64,100,80,84,91,84,28,24,48,72,92,104,124,132,54,56,100,144,88,92,109,64;

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.

Cf. A139251, A194701, A194445, A220501, A220521, A220524, A220527.

a(n) = A221528(n)/4.

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.

cp's OEIS Frontend

A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

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A194441 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194440.

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A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

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A194444 D-toothpick sequence of the second kind in the first quadrant.

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A182635 Number of toothpicks added at n-th stage to the toothpick structure of A182634.

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A220523 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "narrow" triangle of A220522.

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

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A220525 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220524.

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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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