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 1-4 of 4 results.

A161413 First differences of A161412.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 3, 4, 6, 4, 5, 5, 10, 9, 5
Offset: 1

Views

Author

Omar E. Pol, Jun 10 2009

Keywords

Comments

Number of V-toothpicks added to the structure at the n-th round.

Links

Crossrefs

Cf. A139251, A160121, A160407, A161207, A161329, A161412.

A161414 Primes in A161412.

Original entry on oeis.org

2, 3, 5, 19, 29
Offset: 1

Views

Author

Omar E. Pol, Jun 10 2009

Keywords

Crossrefs

Cf. A139253, A161412, A161423.

A161420 V-toothpick sequence starting from a V-toothpick whose central point touch a straight line, as a V.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 19, 25, 31, 39, 51, 59, 69, 79, 99, 117, 127, 143, 163, 171
Offset: 0

Views

Author

Omar E. Pol, Jun 10 2009

Keywords

Comments

The sequence gives the number of V-toothpicks after n-th stages on hexagonal net. A161421 (the first differences) gives the number added at the n-th stage. See A161206 for more information.
Also, it appears this is a H-toothpick sequence in the first quadrant on the square grid, starting with a D-toothpick from the point (0,0). The sequence gives the number of toothpicks and D-toothpicks after n-th stage. A161421 (the first differences) gives the number added at the n-th stage. For more information see A182838.

Links

Crossrefs

Cf. A139250, A160120, A160406, A161206, A161328, A161412, A161421, A161422, A161423, A182832, A182838.

A355310 Total number of V-toothpicks after n-th stage in a cellular automaton with V-toothpicks of 60 degrees (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 27, 37, 51, 69, 79, 89, 103, 123, 141, 165, 201, 245, 267
Offset: 0

Views

Author

Jean Hoffmann and Omar E. Pol, Jul 20 2022

Keywords

Comments

An idea from Jean Hoffmann.
In this cellular automaton a V-toothpick is formed by 2 toothpicks of length 1 that share a vertex and the angle between both toothpicks is 60 degrees.
On the infinite triangular grid we start with no V-toothpick, so a(0) = 0.
At stage 1 we place a V-toothpick upside down, so a(1) = 1.
At every stage the V-toothpicks of the new generation must be connected to the structure by touching with their middle vertex the free ends of the V-toothpicks of the previous generation following a special rule:
The new V-toothpicks must be placed between the imaginary straight line containing the two extreme ends of the V-toothpick of the previous generation and the imaginary straight line that contains the middle vertex of that V-toothpick and that it is parallel to the aforementioned straight line.
A355311(n) gives the number of V-toothpicks added to the structure at the n-th stage.
2*a(n) is the total number of toothpicks of length 1 in the structure after n-th stage.
This cellular automaton is a companion of the Y-toothpick cellular automaton of A160120 in the sense that both essentially grow as an equilateral triangle.
This cellular automaton is slightly less symmetrical than Y-toothpick cellular automaton because its structure has a "backbone" formed by concave hexagons from the center of the triangle to one of its vertices.
The behavior could be very close to A160120 and similar to A153006 (see the graph).
After 18 stages we can see in the structure the following polygons:
- Equilateral triangles of perimeter 3.
- Equilateral triangles of perimeter 6 that contain 4 triangular cells.
- Concave hexagons of perimeter 8 that contain 6 triangular cells.
- Concave dodecagons (or concave 12-gons) of perimeter 18 that contain 22 triangular cells.

Examples

			Illustration of initial terms:
.
                                                                  /__\
                                               _\  /_            _\  /_
                                 /__\           /__\            /\/__\/\
            /\      _\/\/_      _\/\/_         _\/\/_          /__\/\/__\
                               /\    /\     _\/\/__\/\/_      _\/\/__\/\/_
                                                             /\          /\
.
  n:         1         2           3              4                 5
  a(n):      1         3           7             13                21
.

Links

Crossrefs

Cf. A355311 (first differences).
Cf. A139250, A153006, A160120, A161206, A161412, A161420, A299476, A299478, A327330, A327332.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.