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.

Showing 1-10 of 12 results. Next

A220501 Number of toothpicks or D-toothpicks added at n-th stage in the structure of A220500.

Original entry on oeis.org

0, 1, 4, 8, 16, 22, 24, 24, 36, 40, 32, 44, 64, 94, 72, 56, 76, 72, 32, 48, 80, 112, 140, 140, 140, 168, 120, 132, 168, 238, 168, 120, 156, 136, 32, 48, 80, 112, 140, 144, 164, 224, 216, 256, 272, 368, 324, 316, 280, 296, 160, 150, 232, 320, 404, 432, 408, 462, 300, 308, 376, 526, 360, 248, 316, 264, 32
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

Essentially the first differences of A220500.

Examples

			Written as an irregular triangle the sequence begins:
0;
1;
4;
8;
16,22;
24,24,36,40;
32,44,64,94,72,56,76,72;
32,48,80,112,140...
.
From _Omar E. Pol_, May 02 2018: (Start)
Also the nonzero terms can be written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 2 as shown below:
1,    4;
8,   16;
22,  24, 24, 36;
40,  32, 44, 64,  94,  72,  56,  76;
72,  32, 48, 80, 112, 140, 140, 140, 168, 120, 132, 168, 238, 168, 120, 156;
136, 32, 48, 80, 112, 140, 144, 164, 224, 216, 256, 272, 368, 324, 316, 280, ...
(End)

Links

Crossrefs

Cf. A047999, A139250, A139251, A194271, A194441, A194443, A194701, A220500, A220521, A220523, A220527.

Extensions

Terms a(23) and beyond from David Applegate's movie version. - Omar E. Pol, May 02 2018

A323651 Number of elements added at n-th stage to the toothpick structure of A323650.

Original entry on oeis.org

1, 2, 4, 8, 4, 8, 12, 24, 4, 8, 12, 24, 12, 24, 36, 72, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216, 12, 24, 36, 72, 36, 72, 108, 216, 36, 72, 108, 216, 108, 216, 324, 648, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216
Offset: 1

Views

Author

Omar E. Pol, Feb 04 2019

Keywords

Comments

The odd-indexed terms (a bisection) gives A147582, the first differences of A147562 (Ulam-Warburton cellular automaton).
The even-indexed terms (a bisection) gives A147582 multiplied by 2.
The word of this cellular automaton is "ab", so 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. See the example.
For further information about the word of cellular automata see A296612.

Examples

			Written as an irregular triangle the sequence begins:
1,2;
4,8;
4,8,12,24;
4,8,12,24,12,24,36,72;
4,8,12,24,12,24,36,72,12,24,36,72,36,72,108,216;
4,8,12,24,12,24,36,72,12,24,36,72,36,72,108,216,12,24,36,72,36,72,108,216,...
...

Links

Crossrefs

First differences of A323650.
Cf. A011782, A139250, A139251, A147562, A147582, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649.
For other hybrid cellular automata, see A194701, A194271, A220501, A290221, A294021, A294981.

Formula

a(2n-1) = A147582(n).
a(2n) = 2*A147582(n).
a(n) = 4*A323641(n-2), n >= 3.

A294963 a(n) is the number of elements added at n-th stage in the structure of the finite cellular automaton of A294962.

Original entry on oeis.org

1, 4, 8, 8, 12, 20, 16, 8, 24, 16
Offset: 1

Views

Author

Omar E. Pol, Feb 10 2018

Keywords

Examples

			The finite sequence can be written as an array of four columns as shown below:
   1,  4,  8, 8;
  12, 20, 16, 8;
  24, 16.
The first column gives the number of toothpicks of length 2.
The second column gives the number of D-toothpicks.
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

Cf. A294962.
Cf. A139251 (toothpicks), A160173 (T-toothpicks), A194701 (D-toothpicks), A220501.
For other hybrid cellular automata, see A289841, A290221, A294021, A294981.

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

Views

Author

Omar E. Pol, Mar 23 2013

Keywords

Comments

Essentially the first differences of A220524.
First differs from A194445 at a(13).

Examples

			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;

Links

Crossrefs

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.
Cf. A139251, A194701, A194445, A220501, A220521, A220524, A220527.

Formula

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

A221528 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220514.

Original entry on oeis.org

0, 4, 8, 16, 16, 16, 32, 44, 32, 16, 32, 64, 96, 64, 80, 100, 64, 16, 32, 64, 96, 128, 144, 168, 176, 80, 96, 176, 256, 160, 176, 212, 128, 16, 32, 64, 96, 128, 144, 176, 208, 168, 192, 256, 400, 320, 336, 364, 336, 112, 96, 192, 288, 368, 416, 496
Offset: 0

Views

Author

Omar E. Pol, Mar 23 2013

Keywords

Comments

Essentially the first differences of A220514.
First differs from A194435 at a(13).

Examples

			When written as an irregular triangle begins:
0;
4;
8;
16,16;
16,32,44,32;
16,32,64,96,64,80,100,64;
16,32,64,96,128,144,168,176,80,96,176,256,160,176,212,128;
16,32,64,96,128,144,176,208,168,192,256,400,320,336,364,336,112,96,192,288,368,416,496,528,216,224,400,576,352,368, 436,256;

Links

Crossrefs

Row lengths give 1 together with A011782.
Cf. A139251, A194435, A194701, A220501, A220521, A220523, A220525.

Formula

a(n) = 4*A220525(n).

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.

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.

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.

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

Views

Author

Omar E. Pol, Dec 23 2012

Keywords

Comments

Essentially the first differences of A220494.
First differs from A194441 (and from A220521) at a(12).

Examples

			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,...

Links

Crossrefs

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