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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A294981 a(n) is the number of elements added at n-th stage to the structure of the cellular automaton of A294980.

Original entry on oeis.org

1, 3, 6, 6, 6, 18, 18, 18, 6
Offset: 1

Views

Author

Omar E. Pol, Feb 10 2018

Keywords

Examples

			This cellular automaton has word "ab". The row lengths are the terms of A011782 multiplied by 2, so the structure of this irregular triangle is as follows:
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;
...
Triangle begins:
1,  3;
6,  6;
6, 18, 18, 18;
...

Links

Crossrefs

Cf. A294980, A296612 (gives more information about the "word" of a cellular automaton).
Cf. A139251, A160121 (Y-toothpicks), A161207 (V-toothpicks).

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

A233971 Number of toothpicks added at n-th stage to the structure of A233970.

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

Views

Author

Omar E. Pol, Dec 18 2013

Keywords

Comments

Essentially the first differences of A233970.
First differs from A170905 at a(24).
First differs from both A233765 and A233781 at a(25).

Examples

			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;
2,4,6,10,10,10,18,24,22,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,22,10,18,28,38,28,46,56,54,8,14,22,26,22,42,56,62,16,30,46,58,32,62,64;
Right border gives A000079.

Crossrefs

Cf. A000079, A011782, A139251, A161207, A161831, A170905, A182633, A182635, A182841, A233761, A233765, A233781, A233970, A233972.

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

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

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.

Examples

			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,...

Crossrefs

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

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

A161421 First differences of A161420.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 8, 6, 6, 8, 12, 8, 10, 10, 20, 18, 10, 16, 20, 8
Offset: 0

Views

Author

Omar E. Pol, Jun 10 2009, Dec 12 2010

Keywords

Comments

Number of V-toothpicks added at the n-th stage to the V-toothpick structure of A161420. See also A161206 and A161207.
Also it appears a(n) is also the number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A161420. See also A182838 and A182839.

Links

Crossrefs

Cf. A139251, A161206, A161207, A161420, A161422, A182633, A182838, A182839.

A173531 a(0)=0: For n>0, a(n) = A060632(n)*A060632(n-1).

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 16, 8, 4, 8, 16, 16, 16, 32, 64, 16, 4, 8, 16, 16, 16, 32, 64, 32, 16, 32, 64, 64, 64, 128, 256, 32, 4, 8, 16, 16, 16, 32, 64, 32, 16, 32, 64, 64, 64, 128, 256, 64, 16, 32, 64, 64, 64, 128, 256, 128, 64, 128, 256, 256
Offset: 0

Views

Author

Omar E. Pol, Oct 10 2010

Keywords

Comments

First differences of A173530.
Number of triangles (Or V-toothpicks, or L-toothpicks, etc.) added in the three-dimensional structure of A173530 at the n-th stage.

Examples

			If written as a triangle, begins:
0;
1;
2;
4,4;
4,8,16,8;
4,8,16,16,16,32,64,16;
4,8,16,16,16,32,64,32,16,32,64,64,64,128,256,32;
4,8,16,16,16,32,64,32,16,32,64,64,64,128,256,64,16,32,64,64,64,128,256,128,...

Links

Crossrefs

Cf. A001316, A047999, A060632, A139250, A139251, A161207, A172311, A173530, A173532.

Programs

  • Mathematica
    Prepend[Times @@@ Partition[Array[2^DigitCount[Floor[#/2], 2, 1] &, 120, 0], 2, 1], 0] (* Michael De Vlieger, Jan 11 2024 *)

A161413 First differences of A161412.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 3, 4, 6, 4, 5, 5, 10, 9, 5
Offset: 1

Views

Author

Omar E. Pol, Jun 10 2009

Keywords

Comments

Number of V-toothpicks added to the structure at the n-th round.

Links

Crossrefs

Cf. A139251, A160121, A160407, A161207, A161329, A161412.

A173532 a(n) = A173531(n) - A139251(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 4, 0, 4, 32, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4, 28, 24, 4, 40, 176, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4, 28, 24, 4, 40, 176, 28, 0, 4, 28, 24, 4, 40, 172, 72, 4, 36, 144, 116, 48, 256, 832, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4
Offset: 0

Views

Author

Omar E. Pol, Oct 10 2010

Keywords

Examples

			If written as a triangle, begins:
0;
0;
0,0;
0,0,0,4;
0,0,0,4,4,0,4,32;
0,0,0,4,4,0,4,32,12,0,4,28,24,4,40,176;
0,0,0,4,4,0,4,32,12,0,4,28,24,4,40,176,28,0,4,28,24,4,40,172,72,4,36,144,116,48,256,832;

Links

Crossrefs

Cf. A139250, A139251, A161207, A172311, A173530, A173531.

A161208 a(n) = (A161206(n+1)-1)/2.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 34, 40, 49, 61, 76, 91, 105, 120, 130, 136, 145, 158, 175, 196, 221, 249, 276, 298, 322, 354, 395, 435, 469, 502, 520, 526, 535, 548, 565, 586, 611, 640, 669, 696, 729, 774, 831, 894, 955, 1015, 1066, 1096, 1120, 1155, 1202, 1261, 1332
Offset: 0

Views

Author

Omar E. Pol, Jun 08 2009

Keywords

Crossrefs

Cf. A161206, A161207, A161209.

Extensions

More terms from Jinyuan Wang, Mar 01 2020
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