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 13 results. Next

A220523 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "narrow" triangle of A220522.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 8, 8, 13, 16, 4, 4, 8, 12, 16, 16, 20, 24, 12, 8, 16, 28, 16, 16, 25, 32, 4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 32, 40, 41, 48, 20, 8
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

Essentially the first differences of A220522. First differs from A194443 at a(47).

Examples

			Written as an irregular triangle begins:
0;
1;
2;
4,4;
4,4,7,8;
4,4,8,12,8,8,13,16;
4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32;
4,4,8,12,16,16,22,32,26,20,24,40,32,40,41,48,20,8,...

Links

Crossrefs

Cf. A139250, A139251, A194441, A194443, A194445, A220521, A220522.

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

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.

A220520 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the third kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 63, 79, 91, 107, 123, 127, 131, 139, 151, 167, 187, 211, 237, 261, 273, 293, 325, 365, 393, 425, 457, 461, 465, 473, 485, 501, 521, 545, 571, 595, 615, 647, 691, 755, 807, 855, 909, 944, 961, 981, 1017, 1065
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

The structure is essentially one of the vertical wedges of several D-toothpick structures. For more information see A220500. First differs from A194440 at a(14). The first differences (A220521) give the number of toothpicks or D-toothpicks added at n-th stage. See A220522 for the "narrow" triangle of the third kind.

Links

Crossrefs

Cf. A047999, A139250, A194270, A194440, A194442, A220494, A220521, A220522.

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).

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.

A220512 D-toothpick sequence of the third kind starting with a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 88, 136, 168, 184, 216, 280, 344, 424, 508, 620, 684, 700, 732, 796, 892, 1004, 1148, 1324, 1460, 1572, 1668, 1844, 2020, 2228, 2424, 2664, 2792, 2808, 2840, 2904, 3000, 3112, 3264
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. A221565 (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 A194432 at a(14).

Links

Crossrefs

Cf. A139250, A194270, A194432, A194700, A220500, A220514, A220520, A220522, A220524, A220526.

A233762 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind, second version.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 20, 28, 30, 34, 40, 52, 58, 64, 75, 91, 93, 97, 103, 115, 129, 145, 163, 187, 197, 205, 219, 247, 261, 277, 300, 332, 334, 338, 344, 356, 370, 386, 406, 438, 464, 484, 506, 546, 576, 616, 655, 703, 721, 729
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2013

Keywords

Comments

The structure is essentially the same as A220522 but here the borders do not contain D-toothpicks with exposed endpoints except the initial D-toothpick. The structure is one of the oblique wedges of several D-toothpick structures. For more information see A220500. The first differences (A233763) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A139250, A220500, A220522, A233760, A233763.

A256940 a(n) is the total number of free ends of a certain configuration of line segments after n iterations (see Comments lines for definition).

Original entry on oeis.org

2, 4, 8, 12, 12, 12, 20, 20, 16, 24, 28, 48, 52, 36, 44, 36, 16, 24, 40, 56, 72, 72, 76, 80, 60, 64, 80, 124, 132, 88, 100, 68, 16, 24, 40, 56, 72, 80, 88, 104, 112, 128, 176, 216, 244, 212, 168, 148, 84, 64, 104, 152, 200, 200, 212, 216, 148, 144, 176, 276, 296, 192, 212, 136, 16
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 19 2015

Keywords

Comments

The initial pattern is a straight line segment which has 2 free ends: a(0)=2.
The construction rules for the following generations are:
(i) add 2 line segments (all line segments are of equal length) at each free end of previous generation by arranging them in a "V" shape at angle Pi/2 and symmetrically placed at the free end,
(ii) overlaps among different generations are prohibited (if, for a given free end, any of the two new segments from its "V" touch or cross a segment from an earlier generation, then the entire "V" is not added, and that free end is just declared non-free),
(iii) the {a(n)} free ends are the ends of elements that do not touch or cross the others (if a new segment is touched by another segment only at the endpoint which it shares with its parent, then this doesn't count as an intersection and its other end is considered free).
It seems that a(n) drop to 16 for n = 8, 16, 32, 64,... . See illustration in the links.
The structure of the illustration of initial terms is very similar to the structure of A194270 and A220500. - Omar E. Pol, Apr 19 2015

Links

Crossrefs

Cf. A194270, A194440, A194442, A220500, A220520, A220522, A256641, A256941.

Programs

  • Mathematica
    new2[{{s_, t_}, a_}] := Simplify@Table[{{t, AngleVector[t, {1, a + si Pi/4}]}, a + si Pi/4}, {si, {1, -1}}];
xx[l1_, l2_] := SquaredEuclideanDistance[First@l1, First@l2] <= 4 && With[{int = Simplify@RegionIntersection[Line@l1, Line@l2]}, int =!= EmptyRegion[2] && int =!= Point[{First@l2}] && int =!= Point[{First@l1}]];
{nonfree, free} = {{}, {{{{1/2, 0}, {1, 0}}, 0}, {{{1/2, 0}, {0, 0}}, Pi}}};
a = {2};
next[] := ({oldnonfree, oldfree, nonfree, free} = {nonfree, free, Join[free, nonfree], {}};
  Do[n2 = new2[f]; If[And @@ Table[AllTrue[oldnonfree, ! xx[First@#, First@new] &], {new, n2}], Do[
    tt = GroupBy[free, xx[First@#, First@new] &];
    free = Lookup[tt, False, {}];
    If[KeyExistsQ[tt, True], nonfree = Join[nonfree, tt[True], {new}], AppendTo[free, new]];
  , {new, n2}]], {f, oldfree}];
  AppendTo[a, Length@free];);
Do[next[], {10}];
a (* Andrey Zabolotskiy, Mar 09 2025 *)

Extensions

a(1) = 2 prepended and a(3) = 8 corrected by Omar E. Pol, Apr 19 2015
Partially edited by Kival Ngaokrajang, as Omar E. Pol suggestion, Apr 26 2015
Terms a(12), a(13), a(59) corrected by Kival Ngaokrajang, Apr 26 2015
Terms a(27), a(60), a(63) corrected, other terms verified, description clarified by Andrey Zabolotskiy, Mar 09 2025
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