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 37 results. Next

A194271 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194270.

Original entry on oeis.org

0, 1, 4, 8, 16, 22, 24, 22, 40, 40, 32, 32, 56, 74, 96, 50, 88, 72, 32, 48, 72, 104, 128, 112, 144, 144, 152, 96, 152, 178, 240, 122, 184, 136, 32, 48, 72, 108, 144, 144, 184, 188, 200, 176, 272, 274, 416, 250, 288, 272, 216, 144, 208, 292, 384, 332, 376
Offset: 0

Views

Author

Omar E. Pol, Aug 23 2011

Keywords

Comments

Essentially the first differences of A194270.

Examples

			Written as a triangle:
0,
1,
4,
8,
16,22,
24,22,40,40,
32,32,56,74,96,50,88,72,
32,48,72,104,128,112,144,144,152,96,152,178,240,122,184,136,
32,48,72,108,144,144,184,188,200,176,272,274,416,250,288,...

Links

Crossrefs

Cf. A010694, A129370, A139250, A139251, A172311, A182839, A194270, A194441, A194443, A194445.

Formula

a(n) = n^2-(n-1)^2*(1-(-1)^n)/8, if 0 <= n <=4.
Let b(n) = A194441(n), let c(n) = A194443(n), let d(n) = A010694(n), then:
Conjecture: a(n) = 4*(b(n-1)-d(n)) + 2*(c(n)-d(n+1)) + 2*(c(n+2)-d(n+1)) + 8, if n >= 3.
Conjecture: a(2^k+2) = 32, if k >= 3.

Extensions

More terms from Omar E. Pol, Sep 01 2011

A194277 Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.

Original entry on oeis.org

2, 4, 3, 6, 7, 2, 7, 7, 2, 3, 3, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1
Offset: 3

Views

Author

Omar E. Pol, Aug 25 2011

Keywords

Comments

WARNING: The numbers are not fully tested. A new polygonal shape may appear in the structure beyond the stage 128 of A194270.
The cellular automaton of A194270 contains a large number of distinct polygonal shapes. For simplicity we call "polygons" to polygonal shapes.
In order to construct this sequence we use the following rules:
- Consider only the convex polygons and the concave polygons. Self-intersecting polygons are not counted (Note that some polygons contain in their body a toothpick or D-toothpick with an exposed endpoint; that element is not a part of the perimeter of the polygons).
- If two polygons have the same shape but they have different size then these polygons must be counted as distinct types of polygons.
- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct types of polygons.
For more information see A194276 and A194278.

Examples

			Consider toothpicks of length 2 and D-toothpicks of length sqrt(2):
a(3) = 2 because the structure contains 2 types of triangles, each with area: 1, 2.
a(4) = 4 because the structure contains 4 types of quadrilaterals: 3 squares, each with area: 2, 4, 8 and also a rectangle with area 8.
a(5) = 3 because the structure contains 3 types of pentagons: a concave pentagon with area = 3 and also 2 convex pentagons with area 5 and 6.
a(12) = 3 because the structure contains 3 types of dodecagons: a symmetric concave dodecagon with area 29 and also 2 asymmetrict concave dodecagons both with area = 18. These last dodecagons are essentially equal but with reflected shape, so a(12) = 3 not 2.

Links

Crossrefs

Cf. A194270, A194276, A194278, A194444.

A194276 Number of distinct polygonal shapes after n-th stage in the D-toothpick structure of A194270.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 4, 5, 6, 7, 9, 10, 10, 11, 13, 13, 14
Offset: 0

Views

Author

Omar E. Pol, Aug 23 2011

Keywords

Comments

The cellular automaton of A194270 contains a large number of distinct polygonal shapes. For simplicity we also call polygonal shapes "polygons".
In order to construct this sequence we use the following rules:
- Consider only the convex polygons and the concave polygons. Self-intersecting polygons are not counted. (Note that some polygons contain in their body a toothpick or D-toothpick with an exposed endpoint; that element is not a part of the perimeter of the polygon.)
- If two polygons have the same shape but they have different size then these polygons must be counted as distinct polygonal shapes.
- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct polygonal shapes.
For more information see A194277 and A194278.
Question: Is there a maximal record in this sequence?

Examples

			Consider toothpicks of length 2 and D-toothpicks of length sqrt(2).
.
Stage       New type  Perimeter    Area   Term       a(n)
. 0            -          -          -    a(0) =       0
. 1            -          -          -    a(1) =       0
. 2            -          -          -    a(2) =       0
. 3            -          -          -    a(3) =       0
. 4         hexagon   4*sqrt(2)+4    6    a(4) =       1
. 5   5.1   hexagon   2*sqrt(2)+8    8
.     5.2   octagon   4*sqrt(2)+8   14    a(5) = 1+2 = 3
. 6         pentagon  2*sqrt(2)+6    5    a(6) = 3+1 = 4
. 7         enneagon  6*sqrt(2)+6   13    a(7) = 4+1 = 5

Links

Crossrefs

Cf. A139250, A194270, A194271, A194277, A194278, A194444.

A194278 Total number of polygons after n-th stage in the D-toothpick structure of A194270.

Original entry on oeis.org

0, 0, 0, 0, 2, 8, 14, 16, 26, 38, 46, 48, 56, 72, 102
Offset: 0

Views

Author

Omar E. Pol, Aug 26 2011

Keywords

Comments

The structure of the D-toothpick cellular automaton contains at least several tens of different types of polygons. For more information see A194276 and A194277.

Examples

			Consider the structure with toothpicks of length 2 and D-toothpicks of length sqrt(2). After 3 stages the number of polygons in the structure is equal to 0. After 4 stages there are 2 hexagons, each with area = 6, so a(4) = 2. After 5 stages there are new 6 polygons: 2 hexagons, each with area = 8 and also 2 octagons, each with area = 14, so a(5) = 2+6 = 8.

Links

Crossrefs

Cf. A160124, A170926, A161418, A161828, A161836, A194270, A194276, A194277.

A194283 Numbers n such that at stage n of A194270 appears for first time a new distinct polygonal shape in the structure.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 21
Offset: 1

Views

Author

Omar E. Pol, Sep 02 2011

Keywords

Comments

This sequence contains at least 25 terms. The last term is > 200, if this sequence is finite. See also A194277.
For more information about the polygonal shapes in the structure of A194270 see A194276 and A194278.

Links

Crossrefs

Cf. A194270, A194276, A194277, A194278, A194440, A194442

A220500 D-toothpick sequence of the third kind starting with a single toothpick.

Original entry on oeis.org

0, 1, 5, 13, 29, 51, 75, 99, 135, 175, 207, 251, 315, 409, 481, 537, 613, 685, 717, 765, 845, 957, 1097, 1237, 1377, 1545, 1665, 1797, 1965, 2203, 2371, 2491, 2647, 2783, 2815, 2863, 2943, 3055, 3195, 3339, 3503, 3727, 3943, 4199, 4471, 4839, 5163, 5479, 5759, 6055, 6215, 6365, 6597, 6917, 7321, 7753, 8161
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

This is a cellular automaton of forking paths to 135 degrees which uses elements of three sizes: toothpicks of length 1, D-toothpicks of length 2^(1/2) and D-toothpicks of length 2^(1/2)/2. Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
On the infinite square grid we start with no elements.
At stage 1, place a single toothpick on the paper, aligned with the y-axis. The rule for adding new elements is as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. Intersections and overlapping are prohibited.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A220501) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that if n >> 1 the structure looks like an octagon. This C.A. has a fractal (or fractal-like) behavior related to powers of 2. Note that for some values of n we can see an internal growth.
The structure contains eight wedges. Each vertical wedge (see A220520) also contains infinitely many copies of the oblique wedges. Each oblique wedge (see A220522) also contains infinitely many copies of the vertical wedges. Finally, each horizontal wedge also contains infinitely many copies of the vertical wedges and of the oblique wedges.
The structure is mysterious: it contains at least 59 distinct internal regions (or polygonal pieces), for example: one of the concave octagons appears for first time at stage 223. The largest known polygon is a concave 24-gon. The exact number of distinct polygons is unknown.
Also the structure contains infinitely many copies of two subsets of distinct size which are formed by five polygons: three hexagons, a 9-gon and a pentagon. These subsets have a surprising connection with the Sierpinski triangle A047999, but the pattern is more complex.
Apparently this cellular automaton has the most complex structure of all the toothpick structures that have been studied (see illustrationsm also the illustrations of the wedges in the entries A220520 and A220522).
The structure contains at least 69 distinct polygonal pieces. The largest known polygon is a concave 24-gon of area 95/2 = 47.5 which appears for first time at stage 879. - Omar E. Pol, Feb 10 2018

Links

Crossrefs

Cf. A139250, A172310, A194270, A194276, A194432, A194434, A194440, A194442, A194444, A194700, A220520, A220522.

Extensions

Terms a(23) and beyond from David Applegate's movie version. - Omar E. Pol, Feb 10 2018

A172310 L-toothpick sequence (see Comment lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 33, 47, 61, 79, 97, 117, 141, 165, 203, 237, 279, 313, 339, 367, 399, 437, 489, 543, 607, 665, 733, 793, 853, 903, 969, 1039, 1109, 1183, 1233, 1285, 1345, 1399, 1463, 1529, 1613, 1701, 1817, 1923, 2055, 2155, 2291, 2417, 2557, 2663, 2781, 2881, 3003, 3109, 3247, 3361, 3499, 3631, 3783, 3939
Offset: 0

Views

Author

Omar E. Pol, Jan 31 2010

Keywords

Comments

We define an "L-toothpick" to consist of two line segments forming an "L".
There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).
The rule for the n-th stage:
If n is odd then we add the large L-toothpicks to the structure, otherwise we add the small L-toothpicks to the structure.
Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.
The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation.
We start at stage 0 with no L-toothpicks.
At stage 1 we place a large L-toothpick in the horizontal direction, as a "V", anywhere in the plane (Note that there are two exposed endpoints).
At stage 2 we place two small L-toothpicks.
At stage 3 we place four large L-toothpicks.
At stage 4 we place six small L-toothpicks.
And so on...
The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th stage.
For more information see A139250, the toothpick sequence.
In calculating the extension, the "special rule" was strengthened to prohibit intersections as well as overlappings. [From John W. Layman, Feb 04 2010]
Note that the endpoints of the L-toothpicks of the new generation can touch the L-toothpìcks of old generations but the crosses and overlaps are prohibited. - Omar E. Pol, Mar 26 2016
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., A194270). 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 A172304.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172311, A172312, A194270, A220500.

Extensions

Terms a(9)-a(41) from John W. Layman, Feb 04 2010
Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

A194440 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the second kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 63, 79, 87, 103, 119, 123, 127, 135, 147, 163, 179, 203, 229, 253, 265, 285, 313, 353, 373, 405, 437, 441, 445, 453, 465, 481, 497, 521, 547, 571, 591, 623, 663, 727, 767, 815, 869, 909, 921, 941, 973, 1021
Offset: 0

Views

Author

Omar E. Pol, Aug 29 2011

Keywords

Comments

For the D-toothpick "narrow" triangle of the second kind see A194442.
The structure is essentially one of the wedges of several D-toothpick structures. For more information see A194270. The first differences (A194441) give the number of toothpicks or D-toothpicks added at n-th stage. [Omar E. Pol, Dec 29 2012]

Links

Crossrefs

Cf. A139250, A160120, A160406, A161830, A194270, A194277, A194441, A194442, A194444, A194692, A194693, A194696, A220494, A220520.

A194442 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the second kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 693, 741, 761, 769, 785, 813, 853, 897, 947
Offset: 0

Views

Author

Omar E. Pol, Aug 29 2011

Keywords

Comments

If n = 2^k, k >= 1, then the structure looks like an isosceles triangle. For the D-toothpick "wide" triangle of the second kind see A194440.
The structure is essentially one of the wedges of several D-toothpick structures. For more information see A194270. The first differences (A194443) give the number of toothpicks or D-toothpicks added at n-th stage. - Omar E. Pol, Mar 28 2013

Links

Crossrefs

Cf. A047999, A139250, A160120, A160406, A161830, A194270, A194277, A194440, A194443, A194444, A194694, A194695, A194697, A220496, A220522.

A194700 D-toothpick sequence of the third kind (see Comments lines for definition).

Original entry on oeis.org

0, 1, 5, 13, 29, 51, 75, 97, 137, 177, 209, 241, 297, 371, 467, 533
Offset: 0

Views

Author

Omar E. Pol, Sep 02 2011

Keywords

Comments

This is a cellular automaton of forking paths to 135 degrees which uses elements of three sizes: toothpicks of length 1, D-toothpicks of length 2^(1/2) and D-toothpicks of length 2^(1/2)/2. Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
On the infinite square grid we start with no elements.
At stage 1, place a single D-toothpick on the paper, in diagonal direction.
The rule for adding new elements is as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. Intersections and overlapping are prohibited.
This toothpick structure is very similar to A194270, but here there are no toothpicks endpoints that remain exposed forever.

Links

Crossrefs

Cf. A139250, A194270.
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