A220501 -id:A220501 - cp's OEIS frontend

A299771 a(n) is the number of elements added at n-th stage in the structure of the finite cellular automaton of A299770.

1, 4, 8, 8, 12, 16, 16, 8, 24, 8

Offset: 1

Omar E. Pol, Mar 20 2018

The word of this cellular automaton is abcd. For more information see A296612.

			The finite sequence can be written as an array of four columns as shown below:
   1,  4,  8, 8;
  12, 16, 16, 8;
  24,  8.
The first column gives the number of toothpicks of length 2.
The second column gives the number of D-toothpicks of length sqrt(2).
The third column gives the number of toothpicks of length 1.
The fourth column gives the number of T-toothpicks.
The sequence contains exactly 10 terms.

Very similar to A294963.
Cf. A299770, A296612.
Cf. A139251 (toothpicks), A160173 (T-toothpicks), A194701 (D-toothpicks), A220501.
For other hybrid cellular automata, see A289841, A290221, A294021, A294981.

A323647 Number of elements added at n-th stage to the toothpick structure of A323646.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 6, 12, 14, 12, 6, 12, 14, 16, 18, 32, 34, 20, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 40, 18, 32, 38, 44, 62, 92, 86, 60, 62, 96, 114, 144, 210, 260, 194, 68, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92

Offset: 1

Omar E. Pol, Mar 07 2019

The "word" of this cellular automaton is "ab", but note that this triangle has an unusual structure: an additional row of length 2. For more information about the word of cellular automata see A296612.
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,a,b;
...
Row lengths give 2 together with the terms of A011782 multiplied by 2, also 2 togheter with the column 2 of A296612.
Columns "a" contain numbers of toothpicks of length 2.
Columns "b" contain numbers of D-toothpicks of length 2*sqrt(2). See the example.

			Triangle begins:
1, 2;
2, 4;
6, 6;
6,12,14,12;
6,12,14,16,18,32,34,20;
6,12,14,16,18,32,34,24,18,32,38,44,62,92,82,36;
6,12,14,16,18,32,34,24,18,32,38,44,62,92,82,40,18,32,38,44,62,92,86,60,62,96, ...

First differences of A323646.
Also, 1 together with A160731.
Column 1 gives A134201.
Cf. A011782, A139251, A296612.
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323651, A327331, A327333.

A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 10, 8, 4, 8, 10, 12, 14, 22, 22, 16, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 36, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40, 18, 28, 34, 46, 50, 58, 50, 48, 40, 68, 76, 84, 108, 156, 100, 76, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40

Offset: 1

Omar E. Pol, Sep 01 2019

The word of this cellular automaton is "ab".
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 I-toothpicks.
Columns "b" contain numbers of V-toothpicks.
For further information about the word of cellular automata see A296612.

			Triangle begins:
1,2;
4,4;
4,8,10,8;
4,8,10,12,14,22,22,16;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,36;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,40,18,28,34,46,50,58,50,48,40,68,...

First differences of A327330.
Column 1 gives A123932.
First differs from A231348 at a(11).
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327333 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

A327333 Number of elements added at n-th stage to the toothpick structure of A327332.

Original entry on oeis.org

1, 2, 4, 4, 4, 6, 12, 8, 4, 6, 12, 12, 10, 16, 32, 16, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 32, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 40, 16, 24, 48, 44, 24, 40, 80, 48, 32, 48, 96, 96, 64, 104, 208, 64, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 40

Offset: 1

Omar E. Pol, Sep 01 2019

The word of this cellular automaton is "ab".
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 V-toothpicks. Columns "b" contain numbers of I-toothpicks. See the example.
For further information about the word of cellular automata see A296612.

			Triangle begins:
1,2;
4,4;
4,6,12,8;
4,6,12,12,10,16,32,16;
4,6,12,12,10,16,32,20,12,18,36,36,26,42,84,32;
4,6,12,12,10,16,32,20,12,18,36,36,26,42,84,40,16,24,48,44,24,40,80,48,32,48,...
It appears that right border gives the even powers of 2.

First differences of A327332.
Column 1 gives A123932.
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327331 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

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.

A233761 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "wide" triangle of A233760.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 12, 14, 12, 14, 16, 2, 4, 6, 12, 14, 20, 22, 26, 22, 12, 18, 32, 38, 28, 30, 32, 2, 4, 6, 12, 14, 20, 22, 26, 22, 20, 30, 44, 62, 52, 46, 54, 38, 12, 18, 36, 46, 56, 62, 74, 74, 30, 42, 72, 86, 60, 62, 64, 2, 4, 6, 12

Offset: 0

Omar E. Pol, Dec 16 2013

Essentially the first differences of A233760.

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,12,14,12,14,16;
2,4,6,12,14,20,22,26,22,12,18,32,38,28,30,32;
2,4,6,12,14,20,22,26,22,20,30,44,62,52,46,54,38,12,18,36,46,56,62,74,74,30,42,72,86,60,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A220501, A220521, A233760, A233763, A233781.

a(2n) = A220521(2n); a(2n+1) = A220521(2n+1) - 2.

A233765 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the toothpick "wide" triangle of A233764.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 24, 18, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 18, 10, 18, 28, 38, 30, 50, 54, 34, 8, 14, 22, 26, 22, 42, 56, 50, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10

Offset: 0

Omar E. Pol, Dec 16 2013

Essentially the first differences of A233764.
First differs from A170905 at a(24).
First differs from A233971 at a(25).
First differs from A233781 at a(44).

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,24,18,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,18,10,18,28,38,30,50,54,34,8,14,22,26,22,42,56,50,16,30,46,58,32,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A220501, A220521, A231349, A233764, A233781, A233971.

A221565 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220512.

Original entry on oeis.org

0, 4, 8, 16, 16, 16, 28, 48, 32, 16, 32, 64, 64, 80, 84, 112, 64, 16, 32, 64, 96, 112, 144, 176, 136, 112, 96, 176, 176, 208, 196, 240, 128, 16, 32, 64, 96, 112, 152

Offset: 0

Omar E. Pol, May 13 2013

The first differences of A220512.

First differs from A194433 at a(14).

			When written as a irregular triangle begins:
0;
4;
8;
16,16;
16,28,48,32;
16,32,64,64,80,84,112,64;
16,32,64,96,112,144,176,136,112,96,176,176,208,196,240,128;
16,32,64,96,112,152...

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

Cf. A139251, A194271, A194701, A194443, A220501, A220512, A220521, A220523, A220525, A220527, A221528.

A233763 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "narrow" triangle of A233762.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 5, 8, 2, 4, 6, 12, 6, 8, 11, 16, 2, 4, 6, 12, 14, 16, 18, 24, 10, 8, 14, 28, 14, 16, 23, 32, 2, 4, 6, 12, 14, 16, 20, 32, 24, 20, 22, 40, 30, 40, 39, 48, 18, 8

Offset: 0

Omar E. Pol, Dec 15 2013

Essentially the first differences of A233762.

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,5,8;
2,4,6,12,6,8,11,16;
2,4,6,12,14,16,18,24,10,8,14,28,14,16,23,32;
2,4,6,12,14,16,20,32,24,20,22,40,30,40,39,48,18,8,...
Right border gives A000079.

Cf. A000079, A011782, A139251, A220501, A220523, A233761, A233762.

a(2n) = A220523(2n); a(2n+1) = A220523(2n+1) - 2.

A233782 a(n) = A233781(n+1)/2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 12, 9, 4, 7, 11, 13, 8, 15, 16, 1, 2, 3, 5, 5, 5, 9, 12, 9, 5, 9, 15, 19, 13, 21, 26, 17, 4, 7, 11, 13, 11, 21, 28, 25, 8, 15, 23, 29, 16, 31, 32, 1, 2, 3, 5, 5, 5, 9, 12, 9, 5, 9, 15, 19

Offset: 1

Omar E. Pol, Dec 17 2013

First 22 terms coincide with a(2)-a(23) of A169778.

			Written as an irregular triangle in which the row lengths is A000079 the sequence begins:
1;
1,2;
1,2,3,4;
1,2,3,5,5,4,7,8;
1,2,3,5,5,5,9,12,9,4,7,11,13,8,15,16;
1,2,3,5,5,5,9,12,9,5,9,15,19,13,21,26,17,4,7,11,13,11,21,28,25,8,15,23,29,16,31,32;

Right border gives A000079.

Cf. A139251, A169778, A220501, A220521, A231351, A233781, A233780.

cp's OEIS Frontend

A299771 a(n) is the number of elements added at n-th stage in the structure of the finite cellular automaton of A299770.

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A323647 Number of elements added at n-th stage to the toothpick structure of A323646.

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A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

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A327333 Number of elements added at n-th stage to the toothpick structure of A327332.

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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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A233761 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "wide" triangle of A233760.

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A233765 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the toothpick "wide" triangle of A233764.

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A221565 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220512.

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A233763 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "narrow" triangle of A233762.

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A233782 a(n) = A233781(n+1)/2.

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