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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A220514 D-toothpick sequence of the third 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, 440, 520, 620, 684, 700, 732, 796, 892, 1020, 1164, 1332, 1508, 1588, 1684, 1860, 2116, 2276, 2452, 2664, 2792, 2808, 2840, 2904, 3000, 3128, 3272, 3448, 3656, 3824, 4016, 4272, 4676, 4992
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

This is a toothpick sequence of forking paths to 135 degrees. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A221528 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure has a fractal (or fractal-like) behavior. For more information see A194700.
First differs from A194434 at a(13).

Links

Crossrefs

Cf. A139250, A194270, A194700, A194434, A220500, A220520, A220522, A221528.

Formula

a(n) = 4*A220524(n).

A194692 Corner sequence associated with A194440 and A194441.

Original entry on oeis.org

0, 2, 4, 8, 14, 22, 30, 42, 55, 67, 77, 93, 113, 145, 165, 189, 216, 236, 246, 262, 286, 322, 363, 411, 459, 513, 547, 591, 641, 721, 769, 817, 872, 908, 918, 934, 958, 994, 1035, 1083, 1137, 1205, 1267, 1347, 1427, 1551, 1646, 1750, 1839, 1933, 1977, 2025, 2093
Offset: 0

Views

Author

Omar E. Pol, Sep 01 2011

Keywords

Comments

The sequence gives the number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "corner" structure related to the D-toothpick "wide" triangle (See A194440). The first differences (A194693) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A153006, A170883, A194270, A194440, A194441, A194694.

A194694 Corner sequence associated with A194442 and A194443.

Original entry on oeis.org

0, 2, 4, 8, 14, 22, 30, 41, 57, 70, 80, 92, 112, 134, 160, 184, 220, 241, 251, 263
Offset: 0

Views

Author

Omar E. Pol, Sep 01 2011

Keywords

Comments

The sequence gives the number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "corner" structure related to the D-toothpick "narrow" triangle (See A194442). The first differences (A194695) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A153006, A170883, A194270, A194442, A194443, A194692

A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

Original entry on oeis.org

1, 5, 13, 21, 33, 49, 65, 73, 97, 105
Offset: 1

Views

Author

Omar E. Pol, Mar 20 2018

Keywords

Comments

The structure is essentially the same as the finite structure described in A294962 but here there are no D-toothpicks of length sqrt(2)/2. All D-toothpicks in the structure have length sqrt(2).
The same as A294962, it seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints (or free ends), so the structure is finished.
A299771(n) gives the number of elements added to the structure at n-th stage.
The "word" of this cellular automaton is "abcd". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

Links

Crossrefs

Very similar to A294962.
Cf. A299771, A296612.
Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500.
For other hybrid cellular automata, see A194270, A220500, A289840, A290220, A294020, A294980.

A194696 a(n) = 2*A194693(n).

Original entry on oeis.org

4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 20, 32, 48, 72, 82, 96, 96, 108, 68, 88, 100, 160, 96, 96, 110, 72, 20, 32, 48, 72, 82, 96, 108, 136, 124, 160, 160, 248, 190, 208, 178, 188, 88, 96, 136
Offset: 1

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

Conjecture: number of toothpicks or D-toothpicks added to the structure of A194440 at stage 2^k+n, if k tends to infinity. It appears that rows of A194441 when written as a triangle converge to this sequence.

Examples

			Written as a triangle:
4,
4,
8,
12,16,
16,24,26,24,
20,32,40,64,40,48,54,40,
20,32,48,72,82,96,96,108,68,88,100,160,96,96,110,72,
20,32,48,72,82,96,108,136,124,160,160,248,190,208,178,...

Links

Crossrefs

Cf. A152980, A194270, A194440, A194692, A194693, A194697.

A194697 a(n) = 2*A194695(n).

Original entry on oeis.org

4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 44, 52, 48, 72, 42, 20, 24
Offset: 1

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

Conjecture: number of toothpicks or D-toothpicks added to the structure of A194442 at stage 2^k+n, if k tends to infinity. It appears that rows of A194443 when written as a triangle converge to this sequence.

Examples

			Written as a triangle:
4,
4,
8,
12,16,
16,22,32,26,
20,24,40,44,52,48,72,42,
20,24

Links

Crossrefs

Cf. A152980, A194270, A194442, A194694, A194695, A194696

A212008 D-toothpick sequence of the second kind starting with a single toothpick.

Original entry on oeis.org

0, 1, 5, 13, 29, 51, 71, 95, 131, 171, 203, 247, 303, 397, 457, 513, 589, 661, 693, 741, 813, 925, 1057, 1197, 1333, 1501, 1613, 1745, 1885, 2123, 2271, 2391, 2547, 2683, 2715, 2763, 2835, 2947, 3079
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

This cellular automaton uses elements of two sizes: toothpicks of length 1 and D-toothpicks of length 2^(1/2). 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. If it is possible, 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, otherwise each exposed endpoint of the elements of the old generation must be touched by an endpoint of an element of the new generation such that the angle between the old element and the 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 (A212009) 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 also contains infinitely many copies of the oblique wedges. Each oblique wedge 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 appears to be a puzzle which contains at least 50 distinct internal regions (or polygonal pieces), and possibly more. Some of them appear for first time after 200 stages. The largest known polygon is a concave 24-gon.
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. The distribution of these subsets have a surprising connection with the Sierpinski triangle A047999, but here the pattern is more complex.
For another version see A220500.

Links

Crossrefs

Cf. A047999, A139250, A194270, A194432, A194434, A194440, A194442, A194444, A220500.

A220494 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "wide" triangle of the first kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 59, 67, 75, 91, 107, 111, 115, 123, 131, 139, 147, 163, 179, 187, 195, 211, 227, 243, 259, 291, 323, 327, 331, 339, 347, 355, 363, 379, 395, 403, 411, 427, 443, 459, 475, 507, 539, 547, 555, 571, 587, 603
Offset: 0

Views

Author

Omar E. Pol, Dec 23 2012

Keywords

Comments

This cellular automaton uses toothpicks of length 1 and D-toothpicks of length 2^(1/2). 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 semi-infinite square grid we start with no elements, so a(0) = 0. At stage 1, we place a single toothpick in vertical direction at (0,0),(0,1), so a(1) = 1. Note that there is only one exposed toothpick endpoint.
The rules for adding new elements are 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. The endpoints of the toothpicks of the old generation that are perpendiculars to the initial toothpick remain exposed forever. Overlapping is prohibited.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220495 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that the structure has fractal behavior related to powers of 2. It appears that this cellular automaton has a surprising connection with the Sierpinski triangle, but here the structure is more complex.
For a similar version see A220496. For other more complex versions see A194440, A220520. First differs from A194440 (and from A220520) at a(12).

Links

Crossrefs

Cf. A047999, A139250, A194270, A194440, A220495, A220496, A220520.

A220496 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "narrow" triangle of the first kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 58, 66, 74, 87, 103, 107, 111, 119, 127, 135, 143, 157, 173, 181, 189, 205, 221, 237, 253, 278, 310, 314, 318, 326, 334, 342, 350, 364, 380, 388, 396, 412, 428, 444, 460, 486, 518, 526, 534, 550, 566, 582
Offset: 0

Views

Author

Omar E. Pol, Dec 23 2012

Keywords

Comments

This cellular automaton uses toothpicks of length 1 and D-toothpicks of length 2^(1/2). 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, in the first quadrant, we start with no elements, so a(0) = 0. At stage 1, we place a D-toothpick at (0,0),(1,1), so a(1) = 1. The rules for adding new elements are 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. The endpoints of the D-toothpicks of the old generation that are perpendiculars to the initial D-toothpick remain exposed forever. Overlapping is prohibited.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220497 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that the structure has fractal behavior related to powers of 2. It appears that this cellular automaton has a surprising connection with the Sierpinski triangle, but here the structure is more complex.
For a similar version see A220494. For other more complex versions see A194442, A220522.
First differs from A194442 (and from A220522) at a(12).

Links

Crossrefs

Cf. A047999, A139250, A194270, A194442, A194700, A220494, A220497, A220522.

A220526 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "medium" triangle of the third kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 76, 88, 103, 119, 123, 127, 135, 147, 163, 183, 207, 233
Offset: 0

Views

Author

Omar E. Pol, Jan 02 2013

Keywords

Comments

The structure is essentially one of the horizontal wedges of A220500. First differs from A194442 (and from A220522) at a(13). A220527 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A139250, A194270, A194440, A194700, A194442, A220500, A220512, A220514, A220520, A220522, A220524, A220527.
Previous Showing 21-30 of 37 results. Next

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