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 1-7 of 7 results.

A231349 Number of triangles added at n-th stage to the structure of A231348.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 10, 8, 4, 8, 12, 16, 10, 20, 22, 16, 4, 8, 12, 16, 12, 24, 28, 32, 10, 20, 28, 40, 22, 44, 46, 32, 4, 8, 12, 16, 12, 24, 28, 32, 12, 24, 32, 48, 28, 56, 60, 64, 10, 20, 28, 40, 28, 56, 64, 80, 22, 44, 60, 88, 46, 92, 94, 64
Offset: 1

Views

Author

Omar E. Pol, Dec 15 2013

Keywords

Comments

First differences of A231348.
Observation: the row sums of the first seven rows coincide with the first seven elements of A199923.
Is A199923 the row sums of this triangle?

Examples

			Written as an irregular triangle in which row lengths is A011782 the sequence begins:
1;
2;
4,4;
4,8,10,8;
4,8,12,16,10,20,22,16;
4,8,12,16,12,24,28,32,10,20,28,40,22,44,46,32;
4,8,12,16,12,24,28,32,12,24,32,48,28,56,60,64,10,20,28,40, 28,56,64,80,22,44,60,88,46,92,94,64;

Links

Crossrefs

Right border gives A000079.
Cf. A001316, A011782, A231348.

A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 509, 547, 573, 615, 667, 701, 709, 723, 745, 771
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2013

Keywords

Comments

The D-toothpicks placed in northwest or northeast direction both are prohibited, except in the substructures in which the symmetry could be broken, so a(44) = 509, not 507. For another version with broken symmetry in some substructures see A233764. See also A231348, a simpler cellular automaton based in triangles which has essentially a similar structure.
A233781 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233764 at a(44).

Links

Crossrefs

Cf. A139250, A169780, A220500, A231348, A233764, A233781, A233970.

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.

A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 10, 8, 4, 8, 10, 12, 14, 22, 22, 16, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 36, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40, 18, 28, 34, 46, 50, 58, 50, 48, 40, 68, 76, 84, 108, 156, 100, 76, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40
Offset: 1

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

The word of this cellular automaton is "ab".
The structure of the irregular triangle is as shown below:
a,b;
a,b;
a,b,a,b;
a,b,a,b,a,b,a,b;
a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of I-toothpicks.
Columns "b" contain numbers of V-toothpicks.
For further information about the word of cellular automata see A296612.

Examples

			Triangle begins:
1,2;
4,4;
4,8,10,8;
4,8,10,12,14,22,22,16;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,36;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,40,18,28,34,46,50,58,50,48,40,68,...

Crossrefs

First differences of A327330.
Column 1 gives A123932.
First differs from A231348 at a(11).
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327333 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

A230981 Decimal expansion of 3/(2^(1/2)).

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 0, 3, 4, 3, 5, 5, 9, 6, 4, 2, 5, 7, 3, 2, 0, 2, 5, 3, 3, 0, 8, 6, 3, 1, 4, 5, 4, 7, 1, 1, 7, 8, 5, 4, 5, 0, 7, 8, 1, 3, 0, 6, 5, 4, 2, 2, 1, 0, 9, 7, 6, 5, 0, 1, 9, 6, 0, 6, 9, 8, 6, 0, 9, 8, 7, 1, 7, 6, 9, 3, 1, 6, 0, 5, 5, 8, 2, 7, 5, 5, 8, 1, 3, 0, 1, 4, 9, 1, 4, 6, 2, 3, 5, 9, 1, 0, 2, 5, 2
Offset: 1

Views

Author

Omar E. Pol, Dec 15 2013

Keywords

Comments

Also height of an isosceles triangle with two sides of length 5^(1/2) and base of length 2^(1/2).

Examples

			2.1213203435596425732025330863145471178545...

Crossrefs

Cf. A002193, A152623, A176218, A231348.

Programs

  • Mathematica
    RealDigits[3/Sqrt[2], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Formula

Equals 3/sqrt(2) = (2^(1/2))*3/2 = A002193*A152623.
Equals A176218 - 2. - R. J. Mathar, May 16 2023

A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 507, 545, 575, 625, 679, 713, 721, 735, 757, 783
Offset: 0

Views

Author

Omar E. Pol, Dec 16 2013

Keywords

Comments

The D-toothpicks placed in northwest or northeast direction both are prohibited. Note that due this rule there are substructures with broken symmetry, for instance a(44) = 507, not 509. For another version without broken symmetry see A233780.
A233765 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233780 at a(44).

Links

Crossrefs

Cf. A139250, A169780, A220500, A220520, A231348, A233765, A233780, A233970.

A231351 a(n) = A231349(n+1)/2.

Original entry on oeis.org

1, 2, 2, 2, 4, 5, 4, 2, 4, 6, 8, 5, 10, 11, 8, 2, 4, 6, 8, 6, 12, 14, 16, 5, 10, 14, 20, 11, 22, 23, 16, 2, 4, 6, 8, 6, 12, 14, 16, 6, 12, 16, 24, 14, 28, 30, 32, 5, 10, 14, 20, 14, 28, 32, 40, 11, 22, 30, 44, 23, 46, 47, 32, 2, 4, 6, 8, 6, 12, 14, 16
Offset: 1

Views

Author

Omar E. Pol, Dec 17 2013

Keywords

Comments

Observation: the row sums of the first six rows coincide with the first six elements of A006234.
Is A006234 the row sums of this triangle?

Examples

			Written as an irregular triangle in which row lengths is A000079 the sequence begins:
1;
2,2;
2,4,5,4;
2,4,6,8,5,10,11,8;
2,4,6,8,6,12,14,16,5,10,14,20,11,22,23,16;
2,4,6,8,6,12,14,16,6,12,16,24,14,28,30,32,5,10,14,20,14,28,32,40,11,22,30,44,23,46,47,32;

Links

Crossrefs

Right border gives A000079.
Cf. A006234, A231348, A231349.
Showing 1-7 of 7 results.

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