cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A299476 Toothpick sequence on triangular grid with word "abcb".

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 27, 39, 49, 57, 69, 87, 111, 135, 155, 185, 203, 211, 223, 241, 265, 293, 321, 367, 405, 453, 499, 567, 633, 701, 751, 823, 859, 869, 881, 899, 923, 951, 979, 1025, 1067, 1119, 1179, 1259, 1347, 1439, 1529, 1647, 1731, 1805
Offset: 0

Views

Author

Omar E. Pol, Feb 11 2018

Keywords

Comments

a(n) gives the number of toothpicks of length 2 in the structure after n-th stage.
A299477 (the first differences) gives the number of toothpicks added at n-th stage.
For more information about the "word" of a cellular automaton see A296612.

Links

Crossrefs

Cf. A299477 (first differences), A296612.
Cf. A160120 (word "a"), A139250 (word "ab"), A296510 (word "abc"), A299478 (word "abcbc").

A299478 Toothpick sequence on triangular grid with word "abcbc".

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 27, 33, 43, 59, 79, 91, 101, 117, 137, 169, 193, 215, 249, 285, 325, 341, 355, 375, 399, 439, 471, 503, 547, 595, 659, 707, 749, 807, 877, 981, 1055, 1115, 1193, 1271, 1351, 1375, 1389, 1409, 1433, 1473, 1509, 1549, 1599, 1651, 1723
Offset: 0

Views

Author

Omar E. Pol, Feb 11 2018

Keywords

Comments

a(n) gives the number of toothpicks of length 2 in the structure after n-th stage.
A299479 (the first differences) gives the number of toothpicks added at n-th stage.
For more information about the "word" of a cellular automaton see A296612.

Links

Crossrefs

Cf. A299479 (first differences).
Cf. A160120 (word "a"), A139250 (word "ab"), A296510 (word "abc"), A299476 (word "abcb").

A299477 Number of toothpicks added at n-th stage to the structure of the cellular automaton of A299476.

Original entry on oeis.org

1, 2, 4, 6, 8, 6, 12, 10, 8, 12, 18, 24, 24, 20, 30, 18, 8, 12, 18, 24, 28, 28, 46, 38, 48, 46, 68, 66, 68, 50, 72, 36, 10, 12, 18, 24, 28, 28, 46, 42, 52, 60, 80, 88, 92, 90, 118, 84, 74
Offset: 1

Views

Author

Omar E. Pol, Feb 11 2018

Keywords

Comments

The "word" of this cellular automaton is abcb.
The associated sound to the animation of this cellular automaton could be [tick, tock, tack, tock], [tick, tock, tack, tock], and so on.
For more information about the "word" of a cellular automaton see A296612.

Examples

			The structure of this irregular triangle is shown below:
   a,  b,  c,  b;
   a,  b,  c,  b;
   a,  b,  c,  b,  a,  b,  c,  b;
   a,  b,  c,  b,  a,  b,  c,  b,  a,  b,  c,  b,  a,  b,  c,  b;
...
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 4, the sequence begins:
   1,  2,  4,  6;
   8,  6, 12, 10;
   8, 12, 18, 24, 24, 20, 30, 18;
   8, 12, 18, 24, 28, 28, 46, 38, 48, 46, 68, 66, 68, 50, 72, 36;
  10, 12, 18, 24, 28, 28, 46, 42, 52, 60, 80, 88,92, 90, 118, 84, 74, ...

Links

Crossrefs

Cf. A139251, A160121, A296511 (word "abc"), A299476, A299479 (word "abcbc").

A299479 Number of toothpicks added at n-th stage to the structure of the cellular automaton of A299478.

Original entry on oeis.org

1, 2, 4, 6, 8, 6, 6, 10, 16, 20, 12, 10, 16, 20, 32, 24, 22, 34, 36, 40, 16, 14, 20, 24, 40, 32, 32, 44, 48, 64, 48, 42, 58, 70, 104, 74, 60, 78, 78, 80, 24, 14, 20, 24, 40, 36, 40, 50, 52, 72
Offset: 1

Views

Author

Omar E. Pol, Feb 11 2018

Keywords

Comments

The "word" of this cellular automaton is abcbc.
The associated sound to the animation of this cellular automaton could be [tick, tock, tack, tock, tack], [tick, tock, tack, tock, tack], and so on.
For more information about the "word" of a cellular automaton see A296612.

Examples

			The structure of this irregular triangle is shown below:
   a,  b,  c,  b,  c;
   a,  b,  c,  b,  c;
   a,  b,  c,  b,  c,  a,  b,  c,  b,  c;
   a,  b,  c,  b,  c,  a,  b,  c,  b,  c,  a,  b,  c,  b,  c,   a,  b,  c,  b,  c;
...
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 5, the sequence begins:
   1,  2,  4,  6,  8;
   6,  6, 10, 16, 20;
  12, 10, 16, 20, 32, 24, 22, 34, 36, 40;
  16, 14, 20, 24, 40, 32, 32, 44, 48, 64, 48, 42, 58, 70, 104, 74, 60, 78, 78, 80;
  24, 14, 20, 24, 40, 36, 40, 50, 52, 72, ...

Links

Crossrefs

Cf. A299478.
Cf. A160121 (word "a"), A139251 (word "ab"), A296511 (word "abc"), A299477 (word "abcb").

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

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

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)

Examples

			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

Links

Crossrefs

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

Formula

a(n) = 4*A194445(n).
Conjecture: a(2^k+1) = 16, if k >= 1.

Extensions

More terms from Omar E. Pol, Mar 23 2013

A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

Original entry on oeis.org

1, 5, 13, 21, 33, 49, 65, 73, 97, 105
Offset: 1

Views

Author

Omar E. Pol, Mar 20 2018

Keywords

Comments

The structure is essentially the same as the finite structure described in A294962 but here there are no D-toothpicks of length sqrt(2)/2. All D-toothpicks in the structure have length sqrt(2).
The same as A294962, it seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints (or free ends), so the structure is finished.
A299771(n) gives the number of elements added to the structure at n-th stage.
The "word" of this cellular automaton is "abcd". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

Links

Crossrefs

Very similar to A294962.
Cf. A299771, A296612.
Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500.
For other hybrid cellular automata, see A194270, A220500, A289840, A290220, A294020, A294980.

A182839 Number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A182838.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 6, 10, 8, 4, 6, 12, 16, 14, 14, 22, 16, 4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32, 4, 6, 12, 16, 16, 20, 32, 36, 24
Offset: 0

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

From Omar E. Pol, Feb 06 2023: (Start)
The "word" of this cellular automaton is "ab".
Apart from the initial zero 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;
...
Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).
Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
For further information about the word of cellular automata see A296612.
It appears that the right border of the irregular triangle gives the even powers of 2. (End)

Examples

			From _Omar E. Pol_, Feb 06 2023: (Start)
The nonzero terms can write as an irregular triangle as shown below:
  1, 2;
  4, 4;
  4, 6, 10, 8;
  4, 6, 12, 16, 14, 14, 22, 16;
  4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32;
  ...
(End)

Links

Crossrefs

First differences of A182838.
Cf. A139250, A139251, A161207, A182633, A182635, A182841.
Cf. A000079, A011782, A296612.

Formula

Conjecture: a(n) = (A182841(n+1) + A010673(n))/4, n >= 2. - Omar E. Pol, Feb 10 2023

Extensions

a(19)-a(41) from Omar E. Pol, Jan 06 2023

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

Original entry on oeis.org

1, 4, 8, 8, 12, 16, 16, 8, 24, 8
Offset: 1

Views

Author

Omar E. Pol, Mar 20 2018

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 21, 27, 39, 53, 65, 71, 83, 97, 113, 131, 163, 197, 217, 223, 235, 249, 265, 283, 315, 349, 373, 391, 423, 461, 505, 567, 659, 741, 777, 783, 795, 809, 825, 843, 875, 909, 933, 951, 983, 1021, 1065, 1127, 1219, 1301, 1341, 1359, 1391, 1429, 1473, 1535, 1627, 1713, 1773, 1835, 1931
Offset: 0

Views

Author

Omar E. Pol, Mar 07 2019

Keywords

Comments

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).
For the construction of the sequence the rules are as follows:
On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.
For the next n generations we have that:
At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.
If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.
If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.
If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.
The sequence gives the number of toothpicks and D-toothpicks after n stages.
A323647 (the first differences) gives the number of elements added at the n-th stage.
Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

Examples

			After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.
Note that angle between both D-toothpicks is 90 degrees.
.
                      *
                    *   *
                  * * * * *
                *           *
              *               *
.
After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.
.
                      *
                    *   *
                  * * * * *
                *           *
          * * * * *       * * * * *
.

Links

Crossrefs

Cf. A139250, A168112, A160730, A296612, A323647.
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

Formula

a(n) = 1 + A160730(n-1), n >= 1.
a(n) = 1 + 2*A168112(n-1), n >= 1.

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

Views

Author

Omar E. Pol, Mar 07 2019

Keywords

Comments

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.

Examples

			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, ...

Links

Crossrefs

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.
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