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 1-4 of 4 results.

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

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

Essentially the first differences of A212008.

Examples

			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,...

Links

Crossrefs

Cf. A139250, A139251, A194271, A194433, A194435, A194441, A194443, A194445, A220501.

Formula

It appears that a(2^k + 2) = 32, if k >= 3.

A194444 D-toothpick sequence of the second kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 106, 126, 151, 167, 171, 179, 195, 219, 247, 283, 325, 369, 389, 413, 453, 517, 549, 593, 646, 678, 682, 690, 706, 730, 758, 794, 838, 890, 932, 980, 1040, 1140, 1208, 1292, 1375, 1459, 1487, 1511, 1555
Offset: 0

Views

Author

Omar E. Pol, Aug 24 2011

Keywords

Comments

This cellular automaton has essentially the same rules as A194270. We start at stage 0 with no toothpicks. At stage 1, we place a D-toothpick of length sqrt(2), in diagonal direction, at (0,0),(1,1). At stage 2, we place two toothpicks of length 1. At stage 3 we place four D-toothpicks. And so on. The toothpicks and D-toothpicks are connected by their endpoints. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194445) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure shows a fractal (or fractal-like) behavior.
First differs from A220524 at a(13). - Omar E. Pol, Mar 23 2013

Links

Crossrefs

Cf. A139250, A172310, A182838, A194270, A194432, A194434, A194440, A194442, A194445, A212008, A220524.

Formula

a(n) = A194434(n)/4. - Omar E. Pol, Oct 15 2011

Extensions

More terms from Omar E. Pol, Mar 23 2013

A194434 D-toothpick sequence of the second kind starting with a X-shaped cross formed by 4 D-toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 92, 136, 168, 184, 216, 280, 376, 424, 504, 604, 668, 684, 716, 780, 876, 988, 1132, 1300, 1476, 1556, 1652, 1812, 2068, 2196, 2372, 2584, 2712, 2728, 2760, 2824, 2920, 3032, 3176, 3352, 3560, 3728, 3920, 4160, 4560, 4832, 5168
Offset: 0

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross as a "X", formed by 4 D-toothpicks of length sqrt(2) and centered at the origin. At stage 2, we place 8 toothpicks of length 1. At stage 3, we place 16 D-toothpicks of length sqrt(2). And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194435) give the number of toothpicks or D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal behavior (or fractal-like behavior) related to power of 2, similar to A194270 and very similar to A194432. The octagonal structure contains a large number of distinct closed polygonal regions. For more information see A194270, A194440 and A194442.
First differs from A220514 at a(13). - Omar E. Pol, Mar 23 2013
Observation: at least for the terms in the Data section the graph also appears to be a companion of the graph of A187210 but that could be different comparing more terms. - Omar E. Pol, Jun 24 2022

Links

Crossrefs

Cf. A139250, A187210, A194270, A194432, A194435, A194440, A194442, A194444, A212008, A220514.

Formula

a(n) = 4*A194444(n).

Extensions

More terms from Omar E. Pol, Mar 23 2013

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
Showing 1-4 of 4 results.

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