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.

Previous Showing 11-18 of 18 results.

A194695 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194694.

Original entry on oeis.org

2, 2, 4, 6, 8, 8, 11, 16, 13, 10, 12, 20, 22, 26, 24, 36, 21, 10, 12
Offset: 1

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

Sequence related to the D-toothpick "narrow" triangle (See A194442 and A194443). First differences of A194694.

Examples

			Written as a triangle:
2,
2,
4,
6,8,
8,11,16,13,
10,12,20,22,26,24,36,21,
10,12

Links

Crossrefs

Cf. A152980, A194271, A194442, A194443, A194693, A194694, A194697

Formula

a(n) = A194697(n)/2.

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.

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.

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

A256941 a(n) is the 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, 16, 24, 28, 32, 32, 24, 32, 48, 60, 64, 68, 72, 48, 24, 32, 56, 88, 120, 120, 120, 104, 76, 80, 120, 140, 144, 148, 152, 80, 24, 32, 56, 88, 128, 168, 224, 256, 256, 212, 216, 232, 244, 224, 240, 188, 92, 80, 144, 232, 296, 296, 296, 256, 180, 176, 264, 300, 304, 308, 312, 144, 24
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 19 2015

Keywords

Comments

All line segments are equal length. The initial pattern is a straight line segment. It has 2 free ends, so a(0)=2. The construction rules for generation n >= 1 are:
(i) subject to rule ii, add 2 line segments at each free end by arranging in a "V" shape with angle Pi/3 and connecting symmetrically to the free end (nearly like a 3-handed clock showing 07:00:25);
(ii) a "V" is not added if either of its segments would cross a line segment drawn in an earlier generation;
(iii) when generation n is complete, each new line segment clearly touches 2 line segments where it was initially attached; the other end of the new line segment counts as being free if the segment does not touch or cross any more line segments.
a(n) is the number of free ends created in generation n.
It seems that a(n) drops to 24 for n = 5, 9, 17, 33, 65, ... . See illustrations in the links.
The terms of this sequence should be checked! - Omar E. Pol, Apr 23 2015

Links

Crossrefs

Cf. A194270, A194440, A194442, A220500, A220520, A220522, A256641, A256940.

Extensions

First term suggested by Omar E. Pol, Apr 23 2015
Author's comments edited by Peter Munn, May 11 2021

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
Previous Showing 11-18 of 18 results.

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