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 22 results. Next

A172311 First differences of A172310.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 14, 18, 18, 20, 24, 24, 38, 34, 42, 34, 26, 28, 32, 38, 52, 54, 64, 58, 68, 60, 60, 50, 66, 70, 70, 74, 50, 52, 60, 54, 64, 66, 84, 88, 116, 106, 132, 100, 136, 126, 140, 106, 118, 100, 122, 106, 138, 114, 138, 132, 152, 156, 176, 158, 190, 166, 158, 154, 98, 88, 132, 82, 124, 94, 112
Offset: 0

Views

Author

Omar E. Pol, Jan 31 2010

Keywords

Comments

Number of L-toothpicks added to the L-toothpick structure of A172310 at the n-th stage.

Links

Crossrefs

Cf. A139250, A139251, A160121, A160171, A160173, A161207, A161329, A172310.

Extensions

More terms from Nathaniel Johnston, Nov 15 2010
Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

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

A171178 Permutation of the natural numbers: 0 together with the partial sums of A171177.

Original entry on oeis.org

0, 2, 1, 6, 3, 5, 4, 12, 7, 11, 8, 10, 9, 20, 13, 19, 14, 18, 15, 17, 16, 30, 21, 29, 22, 28, 23, 27, 24, 26, 25, 42, 31, 41, 32, 40, 33, 39, 34, 38, 35, 37, 36, 56, 43, 55, 44, 54, 45, 53, 46, 52, 47, 51, 48, 50, 49, 72, 57, 71, 58, 70, 59, 69, 60, 68
Offset: 0

Views

Author

Omar E. Pol, Feb 23 2010

Keywords

Comments

a(n) is also the value of "x" and "y" of the n-th point (x,y), located on the infinite straight line (0,0),(1,1)..., that is intercepted by the path in structure of A171166.
For another version see A171175.

Links

Crossrefs

Cf. A000027, A005132, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171176, A171178, A172310.

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

A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 5, 3, 4, 4, 5, 11, 13, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17
Offset: 1

Views

Author

Omar E. Pol, Mar 24 2012

Keywords

Comments

Consider a toothpick structure formed by L-toothpicks connected by their endpoints. The endpoints of the L-toothpicks are placed on the main diagonal of the first quadrant. At stage 1 we place an L-toothpick with one of its endpoints on the origin. At stage n we place an L-toothpick of size n. The L-toothpicks are placed alternately, on one or another sector of the first quadrant, trying to make the structure have an exposed endpoint closest to the origin. The total length of all L-toothpicks after the n-th stage is A002378(n). The value of x and y of the endpoint of the structure after the n-th stage is equal to the n-th term of Recamán's sequence A005132(n). Note that we can get other illustrations of initial terms of Recamán's sequence by replacing each L-toothpick by a Q-toothpick or by a semicircumference. This structure is also one of the three views of the three-dimensional model for Recamán's sequence. For more information about L-toothpicks and Q-toothpicks, see A172310 and A187210.

Examples

			The summands are the size of the L-toothpicks:
a(1) = 1.
a(2) = 1 + 2 = 3.
a(3) = 2 + 3 = 5.
a(4) = 3.
a(5) = 4.
a(6) = 4.
a(7) = 5.
a(8) = 5 + 6 = 11.
a(9) = 6 + 7 = 13.
a(10) = 7.

Links

Crossrefs

First differences of A210607.
Cf. A005132, A064289, A119632, A139250, A160356, A160357, A171175, A171178, A172310, A187210, A210604, A210605, A210608-A210613.

A171173 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 positive integers.

Original entry on oeis.org

2, 1, 4, 1, 2, 3, 7, 1, 2, 3, 4, 5, 10, 1, 2, 3, 4, 5, 6, 7, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 1
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2010

Keywords

Comments

The same as A171172 except the initial term.
Also, a(n) is the length of each component of the n-th L-toothpick added to the structure of A171165.
See also A171175, a permutation of the natural numbers.

Examples

			Triangle begins:
2,1,
4,1,2,3,
7,1,2,3,4,5,
10,1,2,3,4,5,6,7,
13,1,2,3,4,5,6,7,8,9,
16,1,2,3,4,5,6,7,8,9,10,11,
19,1,2,3,4,5,6,7,8,9,10,11,12,13,
22,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
25,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,

Links

Crossrefs

Cf. A000027, A033627, A172310, A171164, A161165, A161166, A171172, A171174, A171175, A171176, A171177, A171178.

Programs

  • Mathematica
    Join[{2,1},Flatten[Table[Flatten[{3n+1,Range[2n+1]}],{n,10}]]] (* Harvey P. Dale, Nov 24 2011 *)

A171176 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 positive integers, in reverse order.

Original entry on oeis.org

2, 1, 5, 3, 2, 1, 8, 5, 4, 3, 2, 1, 11, 7, 6, 5, 4, 3, 2, 1, 14, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2010

Keywords

Comments

a(n) is also the length of the n-th L-toothpick added to the structure of A171166.

Examples

			Triangle begins:
   2,  1;
   5,  3,  2,  1;
   8,  5,  4,  3,  2,  1;
  11,  7,  6,  5,  4,  3,  2,  1;
  14,  9,  8,  7,  6,  5,  4,  3,  2, 1;
  17, 11, 10,  9,  8,  7,  6,  5,  4, 3, 2, 1;
  20, 13, 12, 11, 10,  9,  8,  7,  6, 5, 4, 3, 2, 1;
  23, 15, 14, 13, 12, 11, 10,  9,  8, 7, 6, 5, 4, 3, 2, 1;
  26, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;

Links

Crossrefs

Cf. A000027, A016789, A172310, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171177, A171178.
Cf. A132209 (row sums). - R. J. Mathar, Mar 01 2010

Programs

  • Mathematica
    Table[{3n-1,Reverse[Range[2n-1]]},{n,10}]//Flatten (* Harvey P. Dale, Jun 26 2022 *)

A171174 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 numbers <> 0 of A038608.

Original entry on oeis.org

2, -1, 4, -1, 2, -3, 7, -1, 2, -3, 4, -5, 10, -1, 2, -3, 4, -5, 6, -7, 13, -1, 2, -3, 4, -5, 6, -7, 8, -9, 16, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 22, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 25, -1, 2, -3, 4
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2010

Keywords

Comments

Absolute values give A171173.
Note that the partial sums of this sequence gives A171175, a permutation of the natural numbers.

Examples

			Triangle begins:
2, -1,
4, -1,2,-3,
7, -1,2,-3,4,-5,
10,-1,2,-3,4,-5,6,-7,
13,-1,2,-3,4,-5,6,-7,8,-9,
16,-1,2,-3,4,-5,6,-7,8,-9,10,-11,
19,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,
22,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,
25,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,

Crossrefs

Cf. A033627, A038608, A172310, A171164, A161165, A161166, A171172, A171173, A171175, A171176, A171177, A171178.

A171177 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 numbers <> 0 of A038608, in reverse order.

Original entry on oeis.org

2, -1, 5, -3, 2, -1, 8, -5, 4, -3, 2, -1, 11, -7, 6, -5, 4, -3, 2, -1, 14, -9, 8, -7, 6, -5, 4, -3, 2, -1, 17, -11, 10, -9, 8, -7, 6, -5, 4, -3, 2, -1, 20, -13, 12, -11, 10, -9, 8, -7, 6, -5, 4, -3, 2, -1, 23, -15, 14, -13, 12, -11, 10, -9, 8, -7, 6, -5, 4, -3, 2, -1, 26, -17, 16, -15
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2010

Keywords

Comments

Absolute values give A171176.
Note that the partial sums of this sequence give A171178, a permutation of the natural numbers.

Examples

			Triangle begins:
2, -1;
5, -3, 2, -1;
8, -5, 4, -3, 2, -1;
11,-7, 6, -5, 4, -3, 2, -1;
14,-9, 8, -7, 6, -5, 4, -3, 2, -1;
17,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;
20,-13,12,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;
23,-15,14,-13,12,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;
26,-17,16,-15,14,-13,12,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;

Crossrefs

Cf. A016789, A038608, A172310, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171176, A171178.
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