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 11-15 of 15 results.

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 21, 27, 39, 53, 65, 71, 83, 97, 113, 131, 163, 197, 217, 223, 235, 249, 265, 283, 315, 349, 373, 391, 423, 461, 505, 567, 659, 741, 777, 783, 795, 809, 825, 843, 875, 909, 933, 951, 983, 1021, 1065, 1127, 1219, 1301, 1341, 1359, 1391, 1429, 1473, 1535, 1627, 1713, 1773, 1835, 1931
Offset: 0

Views

Author

Omar E. Pol, Mar 07 2019

Keywords

Comments

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).
For the construction of the sequence the rules are as follows:
On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.
For the next n generations we have that:
At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.
If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.
If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.
If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.
The sequence gives the number of toothpicks and D-toothpicks after n stages.
A323647 (the first differences) gives the number of elements added at the n-th stage.
Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

Examples

			After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.
Note that angle between both D-toothpicks is 90 degrees.
.
                      *
                    *   *
                  * * * * *
                *           *
              *               *
.
After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.
.
                      *
                    *   *
                  * * * * *
                *           *
          * * * * *       * * * * *
.

Links

Crossrefs

Cf. A139250, A168112, A160730, A296612, A323647.
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

Formula

a(n) = 1 + A160730(n-1), n >= 1.
a(n) = 1 + 2*A168112(n-1), n >= 1.

A327330 "Concave pentagon" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 63, 75, 89, 111, 133, 149, 153, 161, 171, 183, 197, 219, 241, 261, 275, 299, 327, 361, 403, 463, 511, 547, 551, 559, 569, 581, 595, 617, 639, 659, 673, 697, 725, 759, 801, 861, 909, 949, 967, 995, 1029, 1075, 1125, 1183, 1233, 1281, 1321, 1389, 1465, 1549, 1657
Offset: 0

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks (A160164) and V-toothpicks (A161206).
The surprising fact is that after 2^k stages the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two quadrilaterals (Q1 and Q2), both with their largest sides in vertical position, as shown below:
.
* *
* * * *
* * * *
* * *
* Q1 * Q2 *
* * * *
* * * *
* * * *
* * * *
* * E * *
* * * *
* * * *
** **
* * * * * * * * * *
.
Note that for n >> 1 both quadrilaterals look like right triangles.
Every polygon has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.
If n is odd then we add I-toothpicks in vertical position (see the example).
a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
A327331 (the first differences) gives the number of elements added at the n-th stage.
2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
The structure is almost identical to the structure of A327332, but a little larger at the upper edge.
The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
For another version, very similar, starting with a V-toothpick, see A327332, which it appears that shares infinitely many terms with this sequence.

Examples

			Illustration of initial terms:
.
               |      /|\     |/|\|
               |       |      | | |
                      / \     |/ \|
                              |   |
n   :  0       1       2        3
a(n):  0       1       3        7
After three generations there are five I-toothpicks and two V-toothpicks in the structure, so a(3) = 5 + 2 = 7 (note that in total there are 2*a(3) = 2*7 = 14 single toothpicks of length 1).

Links

Crossrefs

First differs from A231348 at a(11).
Cf. A047999, A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327331 (first differences), A327332 (another version).
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Formula

Conjecture: a(2^k) = A327332(2^k), k >= 0.

A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 33, 41, 45, 51, 63, 75, 85, 101, 133, 149, 153, 159, 171, 183, 193, 209, 241, 261, 273, 291, 327, 363, 389, 431, 515, 547, 551, 557, 569, 581, 591, 607, 639, 659, 671, 689, 725, 761, 787, 829, 913, 953, 969, 993, 1041, 1085, 1109, 1149, 1229, 1277, 1309, 1357, 1453, 1549, 1613
Offset: 0

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

Another version and very similar to A327330.
This arises from a hybrid cellular automaton on a triangular grid formed of V-toothpicks (A161206) and I-toothpicks (A160164).
After 2^k stages, the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two right triangles (R1 and R2) both with their hypotenuses in vertical position, as shown below:
.
* *
* * * *
* * * *
* * *
* R1 * * R2 *
* * * *
* * * *
* * * *
* * E * *
* * * *
* * * *
** **
* * * * * * * * * *
.
Every triangle has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an V-toothpick, formed of two single toothpicks, with its central vertice directed up, like a gable roof, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a I-toothpick formed of two single toothpicks in vertical position.
If n is odd then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof (see the example).
a(n) gives the total number of V-toothpicks and I-toothpicks in the structure after the n-th stage.
A327333 (the first differences) gives the number of elements added at the n-th stage.
2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
The structure is almost identical to the structure of A327330, but a little smaller.
The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
It appears that A327330 shares infinitely many terms with this sequence.

Examples

			Illustration of initial terms:
.
.             /\     |/\|
.                    |  |
.
n:     0       1       2
a(n):  0       1       3
After two generations there are only one V-toothpick and two I-toothpicks in the structure, so a(2) = 1 + 2 = 3 (note that in total there are 2*a(2)= 2*3 = 6 single toothpicks of length 1).

Links

Crossrefs

Cf. A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327333 (first differences), A327330 (another version).
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Formula

Conjecture: a(2^k) = A327330(2^k), k >= 0.

A220512 D-toothpick sequence of the third kind starting with a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 88, 136, 168, 184, 216, 280, 344, 424, 508, 620, 684, 700, 732, 796, 892, 1004, 1148, 1324, 1460, 1572, 1668, 1844, 2020, 2228, 2424, 2664, 2792, 2808, 2840, 2904, 3000, 3112, 3264
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

This is a toothpick sequence of forking paths to 135 degrees. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A221565 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure has a fractal (or fractal-like) behavior. For more information see A194700.
First differs from A194432 at a(14).

Links

Crossrefs

Cf. A139250, A194270, A194432, A194700, A220500, A220514, A220520, A220522, A220524, A220526.

A220497 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220496.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 8, 8, 8, 13, 16, 4, 4, 8, 8, 8, 8, 14, 16, 8, 8, 16, 16, 16, 16, 25, 32, 4, 4, 8, 8, 8, 8, 14, 16, 8, 8, 16, 16, 16, 16, 26, 32, 8, 8, 16, 16, 16, 16, 28, 32, 16, 16, 32, 32, 32, 32, 49, 64, 4, 4, 8, 8, 8, 8, 14, 16
Offset: 0

Views

Author

Omar E. Pol, Dec 23 2012

Keywords

Comments

Essentially the first differences of A220496.
First differs from A194443 (and from A220523) at a(12).

Examples

			Written as an irregular triangle begins:
0;
1;
2;
4,4;
4,4,7,8;
4,4,8,8,8,8,13,16;
4,4,8,8,8,8,14,16,8,8,16,16,16,16,25,32;
4,4,8,8,8,8,14,16,8,8,16,16,16,16,26,32,8,8,16,16,16,16,28,32,16,16,32,32,32,32,49,64;

Links

Crossrefs

Cf. A139250, A139251, A194271, A194443, A194700, A194701, A220495, A220496, A220523.
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