Search: a194278 -id:a194278
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A194270
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D-toothpick sequence of the second kind (see Comments lines for definition).
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+10
37
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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
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OFFSET
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0,3
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COMMENTS
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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]
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LINKS
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EXAMPLE
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Illustration of initial terms:
. o o
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. o o o
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. o o - o o o - o
. / / \ / \
. o o - o o - o o
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. o o o
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. o o
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. 1 5 13
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CROSSREFS
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Cf. A139250, A160120, A172310, A182838, A194271, A194276, A194277, A194278, A194440, A194441, A194442, A194443, A194444.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A194277
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Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.
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+10
7
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2, 4, 3, 6, 7, 2, 7, 7, 2, 3, 3, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1
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OFFSET
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3,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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STATUS
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approved
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A194276
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Number of distinct polygonal shapes after n-th stage in the D-toothpick structure of A194270.
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+10
6
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0, 0, 0, 0, 1, 3, 4, 5, 6, 7, 9, 10, 10, 11, 13, 13, 14
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OFFSET
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0,6
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COMMENTS
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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.
Question: Is there a maximal record in this sequence?
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LINKS
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EXAMPLE
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Consider toothpicks of length 2 and D-toothpicks of length sqrt(2).
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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
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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A194283
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Numbers n such that at stage n of A194270 appears for first time a new distinct polygonal shape in the structure.
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+10
1
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4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 21
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OFFSET
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1,1
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COMMENTS
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This sequence contains at least 25 terms. The last term is > 200, if this sequence is finite. See also A194277.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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