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.

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A161426 Y-toothpick sequence starting at the outside corner of an infinite triangle-shaped polygon as the sieve of A160120 after 2^k rounds.

Original entry on oeis.org

0, 1, 4, 7, 14, 19, 26, 35, 52, 63, 70
Offset: 0

Views

Author

Omar E. Pol and David Applegate, Jun 20 2009

Keywords

Comments

The sequence gives the number of Y-toothpicks after n rounds. A161427 (the first differences) gives the number added at the n-th round.
See the entries A160120, A139250 and A153006 for more information.

Links

Crossrefs

Cf. A000079, A139250, A153006, A160120, A161427, A161830.

A182836 Toothpick sequence starting at the vertex of the outside corner of an infinite 120-degree wedge on hexagonal net.

Original entry on oeis.org

0, 1, 3, 7, 15, 27, 39, 51, 71, 91, 107
Offset: 0

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

Corner sequence for the toothpick structure on hexagonal net.
The sequence gives the number of toothpicks after n stages. A182837 (the first differences) gives the number added at the n-th stage. For more information see A182632 and A153006.

Examples

			We start at stage 0 with no toothpicks.
At stage 1 we place a single toothpick touching a vertex of the infinite hexagon, in direction to the center of the hexagon, but on the outside corner, so a(1)=1.
At stage 2 we place 2 toothpicks touching the exposed endpoint of the initial toothpick, so a(2)=1+2=3.
At stage 3 we place 4 toothpicks, so a(3)=3+4=7.
At stage 4 we place 8 toothpicks, so a(4)=7+8=15.
At stage 5 we place 12 toothpicks, so a(5)=15+12=27.
After 5 stages the toothpick structure has 5 hexagons and 6 exposed endpoints.

Links

Crossrefs

Cf. A139250, A153006, A182632, A182634, A182837, A182840.

A256536 Corner sequence associated with A151723.

Original entry on oeis.org

1, 4, 9, 18, 27, 36, 53, 78, 95, 104, 121, 150, 187, 220, 261, 318, 351, 360, 377, 406, 443, 480, 533, 618, 703, 752, 793, 866, 967, 1060, 1161, 1286, 1351, 1360, 1377, 1406, 1443, 1480, 1533, 1618, 1703, 1756, 1809, 1902, 2035, 2176, 2325, 2522, 2703, 2784, 2825, 2898, 2999, 3108, 3249, 3470, 3723, 3896, 4013, 4186, 4435, 4672, 4909, 5174, 5303
Offset: 1

Views

Author

Omar E. Pol, Apr 01 2015

Keywords

Comments

Total number of ON cells after n generations in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.
For an animation see "The movie version" in Links section.
Partial sums of A256537.
See also the Formula section in A256537.
Compare A256138.

Links

Crossrefs

Cf. A151723, A153006, A169779, A256138, A256537.

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.

A152999 Primes in toothpick sequence A152998.

Original entry on oeis.org

3, 5, 7, 11, 17, 23, 47, 61, 97, 103, 151, 173, 191, 211, 241, 347, 353, 359, 367, 397, 467, 541, 599, 607, 659, 733, 1109, 1237, 1367, 1439, 1453, 1471, 1663, 2029, 2357, 2399, 2671, 2797, 3373, 3607, 3719, 3911, 4241, 5479, 5501, 5527, 5701, 5741, 5779, 5923
Offset: 1

Views

Author

Omar E. Pol, Dec 23 2008

Keywords

Links

Crossrefs

Cf. A139250, A139251, A139253, A152768, A152968, A152978, A152998, A153000, A153002, A153006, A153009.

Extensions

More terms from Jinyuan Wang, Mar 15 2020

A159785 a(n) = A152980(n)*3.

Original entry on oeis.org

3, 6, 9, 9, 12, 21, 24, 15, 12, 21, 27, 30, 45, 66, 60, 27, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 144, 51, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 147, 66, 45, 69, 84, 105, 156, 195, 168, 129, 159, 222, 273, 366, 504, 528
Offset: 1

Views

Author

Omar E. Pol, May 02 2009

Keywords

Links

Crossrefs

Toothpick sequence: A139250.
Cf. A139251, A147646, A152968, A152978, A152980, A153004, A153006.

Extensions

More terms from 3*A152980(n) by Jinyuan Wang, Mar 14 2020

A161417 First differences of A160416.

Original entry on oeis.org

1, 7, 3, 21, 7, 41, 9, 57, 13
Offset: 1

Views

Author

Omar E. Pol, May 20 2009, Jun 14 2009

Keywords

Links

Crossrefs

Cf. A139250, A139251, A153006, A152980, A160411, A160413, A160415, A160416.
Cf. A160797.

A162779 Rows of A162777 when written as a triangle converge to this sequence.

Original entry on oeis.org

1, 3, 5, 5, 7, 13, 15, 9, 7, 13, 17, 19, 29, 43, 39, 17, 7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 95, 33, 7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 97, 43, 29, 45, 55, 69, 103, 129, 111, 85, 105, 147, 181, 243, 335, 351, 223, 65, 7
Offset: 0

Views

Author

Omar E. Pol, Jul 23 2009

Keywords

Comments

It appears that the right border of triangle gives A083318. - Omar E. Pol, Mar 15 2020

Examples

			From _Omar E. Pol_, Mar 15 2020: (Start)
Written as an irregular triangle in which row lengths give A011782 the sequence begins:
1;
3;
5,  5;
7, 13, 15,  9;
7, 13, 17, 19, 29, 43, 39, 17;
7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 95, 33;
7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 97, 43, 29, 45, 55, ...
(End)

Links

Crossrefs

Cf. A083318, A139250, A153003, A153006, A162777.

Extensions

More terms from Jinyuan Wang, Mar 15 2020

A194811 Number of grid points that are covered after n-th stage of A139250 version "Tree", starting with a(0) = 1 and assuming the toothpicks have length 4, 3, and 2.

Original entry on oeis.org

1, 5, 13, 25, 37, 53, 81, 113, 133, 149, 177, 213, 253, 313, 401, 481, 517, 533, 561, 597, 637, 697, 785, 869, 925, 985, 1077, 1189, 1329, 1537, 1793, 1985, 2053, 2069, 2097, 2133, 2173, 2233, 2321, 2405, 2461, 2521, 2613, 2725, 2865, 3073, 3329, 3525, 3613
Offset: 0

Views

Author

Omar E. Pol, Oct 24 2011

Keywords

Comments

The first differences give A147646.

Links

Crossrefs

Cf. A139250, A152980, A153006, A147646, A159795, A160128, A160420, A160422, A160425, A182618, A194800, A194802.

Formula

a(n) = 1 + 4*A153006(n) = 1 + A159795(n).

A153005 Primes in toothpick sequence A153003.

Original entry on oeis.org

7, 31, 127, 211, 487, 571, 643, 811, 1033, 1249, 1663, 1999, 2131, 2179, 2281, 2347, 2467, 3391, 4801, 5059, 6361, 7759, 8191, 8209, 8713, 8779, 8929, 9187, 9343, 9679, 9931, 10687, 13903, 14947, 19009, 19267, 19423, 25057, 26731, 28879, 33289, 35521
Offset: 1

Views

Author

Omar E. Pol, Jan 02 2009

Keywords

Links

Crossrefs

Cf. A139250, A139253, A152998, A152999, A153000, A153002, A153003, A153006, A153009.

Programs

  • Maple
    read("transforms3") ; a139250 := BFILETOLIST("b139250.txt") ; A139250 := proc(n) global a139250; op(n+1,a139250) ; end: A153003 := proc(n) if n =0 then 0; else 1+3/4*(A139250(n+1)-3) ; fi; end: for n from 0 to 400 do p := A153003(n) ; if isprime(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 13 2009
  • Mathematica
    a139250 = Cases[Import["https://oeis.org/A139250/b139250.txt", "Table"], {, }][[All, 2]];
A139250[n_] := a139250[[n + 1]];
A153003[n_] := If[n == 0, 0, 1 + 3/4*(A139250[n + 1] - 3)];
Reap[Do[p = A153003[n]; If[PrimeQ[p], Sow[p]], {n, 0, 400}]][[2, 1]] (* Jean-François Alcover, Apr 05 2020 *)

Extensions

More terms from R. J. Mathar, Jul 13 2009
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