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 1-3 of 3 results.

A161421 First differences of A161420.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 8, 6, 6, 8, 12, 8, 10, 10, 20, 18, 10, 16, 20, 8
Offset: 0

Views

Author

Omar E. Pol, Jun 10 2009, Dec 12 2010

Keywords

Comments

Number of V-toothpicks added at the n-th stage to the V-toothpick structure of A161420. See also A161206 and A161207.
Also it appears a(n) is also the number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A161420. See also A182838 and A182839.

Links

Crossrefs

Cf. A139251, A161206, A161207, A161420, A161422, A182633, A182838, A182839.

A161423 Primes in A161420.

Original entry on oeis.org

3, 5, 7, 11, 19, 31, 59, 79, 127, 163
Offset: 1

Views

Author

Omar E. Pol, Jun 10 2009

Keywords

Links

Crossrefs

Cf. A139253, A161420, A161421.

Extensions

Error in data corrected by D. S. McNeil, Aug 17 2010
a(10) from Michael S. Branicky, Jan 23 2023

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 1-3 of 3 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.