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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A194270 D-toothpick sequence of the second 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, 517, 605, 677, 709, 757, 829, 933, 1061, 1173, 1317, 1461, 1613, 1709, 1861, 2039, 2279, 2401, 2585, 2721, 2753, 2801, 2873, 2981, 3125, 3269, 3453, 3641, 3841, 4017, 4289, 4563, 4979, 5229
Offset: 0

Views

Author

Omar E. Pol, Aug 23 2011

Keywords

Comments

This is a cellular automaton of forking paths to 135 degrees which 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, we place anywhere a D-toothpick.
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, so some toothpick endpoints can remain exposed forever.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194271) give the number of toothpicks and D-toothpicks added at n-th stage.
If n >> 1 the structure looks like an almost regular octagon. The structure has a fractal-like behavior related to powers of 2 (see formula section in A194271 and A194443). Note that for some values of n we can see an internal growth, similar to A160120. Also there are hidden substructures which have a surprising connection with the Sierpinski triangle. The hidden substructures are displayed more clearly for large values of n without reducing the scale of the drawing. The main "wedges" in the structures are essentially the triangles A194440 and A194442.
Note that this structure is much more complex than A139250 and A160120. The structure contains a large number of distinct polygonal shapes. There are convex polygons and concave polygons, also there are symmetrical and asymmetrical polygons. Several of these polygons are also in the structure of A172310. The number of edges of the known polygons are 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 20, 24. Is not known how many distinct types of polygons there are in the structure if n -> infinite. The sequences related with these polygons are A194276, A194277, A194278 and A194283. Note that the structure is not centered with respect to the axes X, Y. Also, for some polygons the area is not an integer. For symmetric versions of C. A. see A194432 and A194434.
Another representation (Large version): instead toothpicks of length 1 we place toothpicks of length 2. We start with no toothpicks. At stage 1, we place a toothpick of length 2 on the y-axis and centered at the origin. At stage 2 we place four D-toothpicks of length 2^(1/2) = sqrt(2), and so on. In this case the structure is centered with respect to the axes X, Y and the area of the polygons is an integer.
[It appears that a normal toothpick is a line segment of length 1 that is parallel to the x-axis or the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. - N. J. A. Sloane, Feb 06 2023]

Examples

			Illustration of initial terms:
.                                            o   o
.                                             \ /
.                        o                     o     o
.                        |                     |    /
.        o               o - o         o       o - o
.       /               /               \     /     \
.      o           o - o                 o - o       o
.                      |                /    |
.                      o               o     o
.                                           / \
.                                          o   o
.
.       1               5                    13
.

Links

Crossrefs

Cf. A139250, A160120, A172310, A182838, A194271, A194276, A194277, A194278, A194440, A194441, A194442, A194443, A194444.

Extensions

More terms from Omar E. Pol, Sep 01 2011

A190430 Positions of 2 in A190427.

Original entry on oeis.org

3, 6, 8, 11, 16, 19, 21, 24, 29, 32, 37, 40, 42, 45, 50, 53, 55, 58, 61, 63, 66, 71, 74, 76, 79, 84, 87, 92, 95, 97, 100, 105, 108, 110, 113, 116, 118, 121, 126, 129, 131, 134, 139, 142, 144, 147, 150, 152, 155, 160, 163, 165, 168, 173, 176, 181, 184, 186, 189, 194, 197, 199, 202, 205, 207, 210, 215
Offset: 1

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190427.

Links

Crossrefs

Cf. A190427, A190428, A190429.

Programs

  • Mathematica
    r = GoldenRatio; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A194277 *)
Flatten[Position[t, 0]] (* A190428 *)
Flatten[Position[t, 1]] (* A190429 *)
Flatten[Position[t, 2]] (* A190430 *)

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.

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
Showing 1-7 of 7 results.

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