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 18 results. Next

A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

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, 33, 48, 20, 8, 16, 28, 40, 44, 50, 60, 28, 16, 32, 60, 32, 32, 49, 64, 4, 4, 8
Offset: 0

Views

Author

Omar E. Pol, Aug 29 2011

Keywords

Comments

Essentially the first differences of A194442. It appears that the structure of the "narrow" triangle is much more regular about n=2^k, see formula section.

Examples

			If written as a triangle:
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,
4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28...
.
It appears that rows converge to A194697.

Links

Crossrefs

Cf. A139251, A160121, A160407, A161831, A194271, A194441, A194442, A194445, A194694, A194695, A194697.

Formula

Conjectures for n = 2^k+j, if -6<=j<=6:
a(2^k-6) = 2^(k-2), if k >= 3.
a(2^k-5) = 2^(k-1), if k >= 3.
a(2^k-4) = 2^k-4, if k >= 2.
a(2^k-3) = 2^(k-1), if k >= 3.
a(2^k-2) = 2^(k-1), if k >= 2.
a(2^k-1) = 3*2^(k-2)+1, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 3.
a(2^k+4) = 12, if k >= 3.
a(2^k+5) = 16, if k >= 4.
a(2^k+6) = 16, if k >= 4.
End of conjectures.

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

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

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

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.

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.

A194444 D-toothpick sequence of the second kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 106, 126, 151, 167, 171, 179, 195, 219, 247, 283, 325, 369, 389, 413, 453, 517, 549, 593, 646, 678, 682, 690, 706, 730, 758, 794, 838, 890, 932, 980, 1040, 1140, 1208, 1292, 1375, 1459, 1487, 1511, 1555
Offset: 0

Views

Author

Omar E. Pol, Aug 24 2011

Keywords

Comments

This cellular automaton has essentially the same rules as A194270. We start at stage 0 with no toothpicks. At stage 1, we place a D-toothpick of length sqrt(2), in diagonal direction, at (0,0),(1,1). At stage 2, we place two toothpicks of length 1. At stage 3 we place four D-toothpicks. And so on. The toothpicks and D-toothpicks are connected by their endpoints. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194445) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure shows a fractal (or fractal-like) behavior.
First differs from A220524 at a(13). - Omar E. Pol, Mar 23 2013

Links

Crossrefs

Cf. A139250, A172310, A182838, A194270, A194432, A194434, A194440, A194442, A194445, A212008, A220524.

Formula

a(n) = A194434(n)/4. - Omar E. Pol, Oct 15 2011

Extensions

More terms from Omar E. Pol, Mar 23 2013

A220522 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third 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, 701, 749, 769, 777
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

The structure is essentially one of the oblique wedges of several D-toothpick structures. For more information see A220500. First differs from A194442 at a(47). The first differences (A220523) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A047999, A139250, A194270, A194700, A194440, A194442, A194444, A220496, A220500, A220523.

A194434 D-toothpick sequence of the second 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, 424, 504, 604, 668, 684, 716, 780, 876, 988, 1132, 1300, 1476, 1556, 1652, 1812, 2068, 2196, 2372, 2584, 2712, 2728, 2760, 2824, 2920, 3032, 3176, 3352, 3560, 3728, 3920, 4160, 4560, 4832, 5168
Offset: 0

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross as a "X", formed by 4 D-toothpicks of length sqrt(2) and centered at the origin. At stage 2, we place 8 toothpicks of length 1. At stage 3, we place 16 D-toothpicks of length sqrt(2). And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194435) give the number of toothpicks or D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal behavior (or fractal-like behavior) related to power of 2, similar to A194270 and very similar to A194432. The octagonal structure contains a large number of distinct closed polygonal regions. For more information see A194270, A194440 and A194442.
First differs from A220514 at a(13). - Omar E. Pol, Mar 23 2013
Observation: at least for the terms in the Data section the graph also appears to be a companion of the graph of A187210 but that could be different comparing more terms. - Omar E. Pol, Jun 24 2022

Links

Crossrefs

Cf. A139250, A187210, A194270, A194432, A194435, A194440, A194442, A194444, A212008, A220514.

Formula

a(n) = 4*A194444(n).

Extensions

More terms from Omar E. Pol, Mar 23 2013

A194432 D-toothpick sequence starting with a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 88, 136, 168, 184, 216
Offset: 0

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross, centered at the origin, formed by 2 vertical toothpicks and 2 horizontal toothpicks of length 1. At stage 2, we place 8 D-toothpicks of length sqrt(2). At stage 3, we place 16 toothpicks of length 1. And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194433) give the number of toothpicks and D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal (or fractal-like) behavior related to power of 2, similar to A194270 and very similar to A194434. The octagonal structure contains a large number of distinct polygonal shapes. For more information see A194270, A194440 and A194442.

Links

Crossrefs

Cf. A139250, A194270, A194433, A194434, A194440, A194442.
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