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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A172304 L-toothpick sequence starting with two opposite L-toothpicks.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 46, 62, 70, 86, 110, 134, 166, 190, 238, 278, 302, 318, 342, 382, 430, 470, 526, 582, 646, 710, 782, 838, 902, 950, 1030, 1118, 1150, 1182, 1246, 1318, 1382, 1422, 1486, 1566, 1662, 1766, 1910, 2006, 2134, 2254, 2414, 2526, 2622
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Comments

The same as A172310 but starting with two L-toothpicks.
We start at stage 0 with no L-toothpicks.
At stage 1 we place two large L-toothpicks in the horizontal direction, as a "X", anywhere in the plane.
At stage 2 we place four small L-toothpicks.
At stage 3 we add eight more large L-toothpicks.
At stage 4 we add eight more small L-toothpicks.
And so on ...
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A212008). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

Links

Crossrefs

For a similar version see A172310.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172305, A172306, A172307, A172308, A172309, A172311, A172312, A172313, A212008, A220500.

Extensions

Terms beyond a(14) from Yan Sheng Ang, Dec 10 2012

A172312 a(n) = A172311(n+1)/2.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 7, 9, 9, 10, 12, 12, 19, 17, 21, 17, 13, 14, 16, 19, 25, 27, 29, 29, 33, 30, 28, 25, 33, 33, 33, 35, 21, 26, 28, 27, 31, 33, 39, 44
Offset: 1

Views

Author

Omar E. Pol, Jan 31 2010

Keywords

Links

Crossrefs

Cf. A139250, A139251, A172310, A172311.

Extensions

More terms from Nathaniel Johnston, Nov 15 2010

A171172 Triangle read by rows in which row n lists 3n-2 together with the first 2n-1 positive integers.

Original entry on oeis.org

1, 1, 4, 1, 2, 3, 7, 1, 2, 3, 4, 5, 10, 1, 2, 3, 4, 5, 6, 7, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 1
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2010

Keywords

Examples

			Triangle begins:
1,1,
4,1,2,3,
7,1,2,3,4,5,
10,1,2,3,4,5,6,7,
13,1,2,3,4,5,6,7,8,9,
16,1,2,3,4,5,6,7,8,9,10,11,
19,1,2,3,4,5,6,7,8,9,10,11,12,13,
22,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
25,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,

Crossrefs

Cf. A016777, A172310, A171173, A171174, A171175, A171176, A171177, A171178.

A172305 Number of L-toothpicks added to the L-toothpick structure of A172304 at the n-th stage.

Original entry on oeis.org

0, 2, 4, 8, 8, 8, 16, 16, 8, 16, 24, 24, 32, 24, 48, 40, 24, 16, 24, 40, 48, 40, 56, 56, 64, 64, 72, 56, 64, 48, 80, 88, 32, 32, 64, 72, 64, 40, 64, 80, 96, 104, 144, 96, 128, 120, 160, 112, 96, 80, 128, 128, 144, 88, 136, 128, 136, 144, 168, 168, 216, 160, 192, 168, 64, 80, 128
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Links

Crossrefs

Cf. First differences of A172304.
Cf. A139250, A139251, A172304, A172306, A172307, A172308, A172309, A172310, A172311, A172312, A172313.

Extensions

More terms from Colin Barker, Apr 19 2015
a(49)-a(66) from Robert Price, Jun 17 2019

A172306 a(n) = A172304(n)/2.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 31, 35, 43, 55, 67, 83, 95, 119, 139, 151, 159, 171, 191, 215, 235, 263, 291, 323, 355, 391, 419, 451, 475, 515, 559, 575, 591, 623, 659, 691, 711, 743, 783, 831, 883, 955, 1003, 1067, 1127, 1207, 1263, 1311, 1351, 1415, 1479, 1551, 1595, 1663, 1727, 1795, 1867, 1951, 2035, 2143, 2223, 2319, 2403, 2435, 2475, 2539, 2587
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Links

Crossrefs

Cf. A139250, A172304, A172305, A172307, A172308, A172309, A172310, A172311, A172312, A172313.

Extensions

a(14)-a(67) from Robert Price, Jun 17 2019

A172307 a(n) = A172305(n)/2.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 8, 4, 8, 12, 12, 16, 12, 24, 20, 12, 8, 12, 20, 24, 20, 28, 28, 32, 32, 36, 28, 32, 24, 40, 44, 16, 16, 32, 36, 32, 20, 32, 40, 48, 52, 72, 48, 64, 60, 80, 56, 48, 40, 64, 64, 72, 44, 68, 64, 68, 72, 84, 84, 108, 80, 96, 84, 32, 40, 64
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Links

Crossrefs

Cf. A139250, A172304, A172305, A172306, A172308, A172309, A172310, A172311, A172312, A172313.

Extensions

a(14)-a(66) from Robert Price, Jun 17 2019

A172308 L-toothpick sequence in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 17, 21, 27, 33, 41, 47, 59, 69, 75, 79, 85, 95, 107, 117, 131, 145, 161, 177, 195, 209, 225, 237, 257, 279, 287, 295, 311, 329, 345, 355, 371, 391, 415, 441, 477, 501, 533, 563, 603, 631, 655
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Comments

The same as A172310 and A172304, but starting from half L-toothpick in the first quadrant.
Note that if n is odd then we add the small L-toothpicks to the structure, otherwise we add the large L-toothpicks to the structure.
We start at stage 0 with half L-toothpick: A segment from (0,0) to (1,1).
At stage 1 we place a small L-toothpick at the exposed toothpick end.
At stage 2 we place two large L-toothpicks.
At stage 3 we place two small L-toothpicks.
At stage 4 we place two large L-toothpicks.
And so on...
The sequence gives the number of L-toothpicks after n stages. A172309 (the first differences) gives the number of L-toothpicks added at the n-th stage.

Links

Crossrefs

Cf. A139250, A153000, A160120, A160170, A160172, A161206, A161328, A172304, A172305, A172306, A172307, A172309, A172310, A172311, A172312, A172313.

Extensions

a(17)-a(47) from Robert Price, Jun 17 2019

A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 4, 2, 4, 6, 6, 8, 6, 12, 10, 6, 4, 6, 10, 12, 10, 14, 14, 16, 16, 18, 14, 16, 12, 20, 22, 8, 8, 16, 18, 16, 10, 16, 20, 24, 26, 36, 24, 32, 30, 40, 28, 24
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Links

Crossrefs

Cf. First differences of A172308.
Cf. A139250, A139251, A152978, A172304, A172305, A172306, A172307, A172308, A172310, A172311, A172312, A172313.

Formula

a(0) = 0, a(n) = A172305(n+1)/4, for n>=1.

Extensions

More terms from Colin Barker, Apr 19 2015

A172313 a(n) = A172309(n+1)/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 4, 3, 6, 5, 3, 2, 3, 5, 6, 5, 7, 7, 8, 8, 9, 7, 8, 6, 10, 11, 4, 4, 8, 9, 8, 5, 8, 10, 12, 13, 18, 12, 16, 15, 20, 14, 12
Offset: 1

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Links

Crossrefs

Cf. A139250, A172304, A172305, A172306, A172307, A172308, A172309, A172310, A172311, A172312.

Extensions

a(16)-a(46) from Robert Price, Jun 17 2019

A210607 Vertex number of an L-toothpick structure which give Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 20, 25, 36
Offset: 1

Views

Author

Omar E. Pol, Mar 24 2012

Keywords

Comments

For more information see A210606.

Links

Crossrefs

First differences give A210606.
A002378, A005132, A119632, A139250, A160356, A171175, A171178, A172310, A187210, A210607-A210613.
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