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 1-8 of 8 results.

A182842 a(n) = A182841(n+2)/2.

Original entry on oeis.org

2, 4, 7, 8, 7, 12, 19, 16, 7, 12, 23, 32, 27, 28, 43, 32, 7, 12, 23, 32, 31, 40, 63, 72, 43, 28, 55, 84, 79, 72, 99, 64, 7, 12, 23, 32, 31, 40, 63, 72, 47, 40, 71, 112, 119, 112, 143, 152, 75, 28, 55, 84, 91, 108, 163, 204, 151, 88, 131, 204, 207, 180, 219, 128
Offset: 0

Views

Author

Omar E. Pol, Dec 11 2010

Keywords

Examples

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle with row lengths A011782:
2;
4;
7, 8;
7, 12, 19, 16;
7, 12, 23, 32, 27, 28, 43, 32;
7, 12, 23, 32, 31, 40, 63, 72, 43, 28, 55, 84, 79, 72, 99, 64;
7, 12, 23, 32, 31, 40, 63, 72, 47, 40, 71, 112, 119, 112, 143, 152, 75, 28, 55, 84, 91, 108, 163, 204, 151, 88, 131, 204, 207, 180, 219, 128;
The right border gives the even powers of 2, at least up a(2^9-1).
(End)

Links

Crossrefs

Cf. A139250, A139251, A151724, A182633, A182840, A182841.

Extensions

More terms from Olaf Voß, Dec 24 2010
Wiki link added by Olaf Voß, Jan 14 2011

A182840 Toothpick sequence on hexagonal net.

Original entry on oeis.org

0, 1, 5, 13, 27, 43, 57, 81, 119, 151, 165, 189, 235, 299, 353, 409, 495, 559, 573, 597, 643, 707, 769, 849, 975, 1119, 1205, 1261, 1371, 1539, 1697, 1841, 2039, 2167, 2181, 2205, 2251, 2315, 2377, 2457, 2583, 2727, 2821, 2901, 3043, 3267, 3505, 3729, 4015
Offset: 0

Views

Author

Omar E. Pol, Dec 09 2010

Keywords

Comments

Rules:
- Each new toothpick must lie on the hexagonal net such that the toothpick endpoints coincide with two consecutive nodes.
- Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.
The sequence gives the number of toothpicks after n stages. A182841 (the first differences) gives the number added at the n-th stage.
The toothpick structure has polygons in which there are uncovered grid points, the same as A160120 and A161206. For more information see A139250.
Has a behavior similar to A151723, A182632. - Omar E. Pol, Feb 28 2013
From Omar E. Pol, Feb 17 2023: (Start)
Assume that every triangular cell has area 1.
It appears that the structure contains only three types of polygons:
- Regular hexagons of area 6.
- Concave decagons (or concave 10-gons) of area 12.
- Concave dodecagons (or concave 12-gons) of area 18.
There are infinitely many of these polygons.
The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.
For an animation see the movie version in the Links section.
The animation shows the fractal-like behavior the same as in other members of the family of toothpick cellular automata.
The structure has internal growth.
For another version starting from a node see A182632.
For a version of the structure in the first quadrant but on the square grid see A182838. (End)

Examples

			We start at stage 0 with no toothpicks.
At stage 1 we place a toothpick anywhere in the plane (For example, in vertical position). There are two exposed endpoints, so a(1)=1.
At stage 2 we place 4 toothpicks. Two new toothpicks touching each exposed endpoint. So a(2)=1+4=5. There are 4 exposed endpoints.
At stage 3 we place 8 toothpicks. a(3)=5+8=13. The structure has 8 exposed endpoints.
At stage 4 we place 14 toothpicks (Not 16) because there are 4 endpoints that are touched by new 8 toothpicks but there are 4 endpoints that are touched by only 6 new toothpicks (not 8), so a(4)=13+14=27.
After 4 stages the toothpick structure has 4 hexagons and 8 exposed endpoints.

Links

Crossrefs

Cf. A139250, A160120, A161206, A182632, A182634, A182838.

Extensions

More terms from Olaf Voß, Dec 24 2010
Wiki link added by Olaf Voß, Jan 14 2011

A187211 First differences of A187210.

Original entry on oeis.org

0, 1, 4, 7, 12, 22, 20, 22, 40, 54, 40, 22, 40, 54, 56, 70, 120, 134, 72, 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136, 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 152, 70, 120, 150, 168, 246, 360, 342, 232, 246, 376, 454, 568, 838, 1032
Offset: 0

Views

Author

Omar E. Pol, Mar 07 2011

Keywords

Comments

Number of Q-toothpicks added at n-th stage to the Q-toothpick structure of A187210.
For the connection with A139251, the first differences of the toothpick sequence A139250, see the Formula section. - Omar E. Pol, Apr 02 2016

Examples

			Written as an irregular triangle the sequence begins:
0;
1;
4;
7;
12;
22, 20;
22, 40, 54, 40;
22, 40, 54, 56, 70, 120, 134, 72;
22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136;
...
The rows of this triangle tend to A188156.
From _Omar E. Pol_, Apr 02 2016: (Start)
For n = 5 we have that A139251(5-2) = 4, A267699(5-2) = 7 and A267695(5-1) = 7, so a(5) = 2*4 + 7 + 7 = 22.
For n = 10 we have that A139251(10-2) = 8, A267699(10-2) = 20 and A267695(10-1) = 4, so a(10) = 2*8 + 20 + 4 = 40.
(End)
Starting from a(3) = 7 the row lengths of triangle are the terms of A011782. - _Omar E. Pol_, Apr 04 2016

Links

Crossrefs

Cf. A011782, A139250, A139251, A172472, A182841, A187210, A187221, A267695, A267699.

Formula

a(2^n + 2) = 16 + 8(2^(n-1) - 1), n >= 3. [Nathaniel Johnston, Mar 26 2011]
From Omar E. Pol, Apr 02 2016: (Start)
a(n) = floor(sqrt(2*n^3)), if 0<=n<=2 or n=6.
a(n) = 2*A139251(n-2) + A267699(n-2) + A267695(n-1), if 3<=n<=5 or n>=7.
(End)

Extensions

Terms after a(7) from Nathaniel Johnston, Mar 26 2011

A182839 Number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A182838.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 6, 10, 8, 4, 6, 12, 16, 14, 14, 22, 16, 4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32, 4, 6, 12, 16, 16, 20, 32, 36, 24
Offset: 0

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

From Omar E. Pol, Feb 06 2023: (Start)
The "word" of this cellular automaton is "ab".
Apart from the initial zero 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;
...
Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).
Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
For further information about the word of cellular automata see A296612.
It appears that the right border of the irregular triangle gives the even powers of 2. (End)

Examples

			From _Omar E. Pol_, Feb 06 2023: (Start)
The nonzero terms can write as an irregular triangle as shown below:
  1, 2;
  4, 4;
  4, 6, 10, 8;
  4, 6, 12, 16, 14, 14, 22, 16;
  4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32;
  ...
(End)

Links

Crossrefs

First differences of A182838.
Cf. A139250, A139251, A161207, A182633, A182635, A182841.
Cf. A000079, A011782, A296612.

Formula

Conjecture: a(n) = (A182841(n+1) + A010673(n))/4, n >= 2. - Omar E. Pol, Feb 10 2023

Extensions

a(19)-a(41) from Omar E. Pol, Jan 06 2023

A222181 Number of pentagons added at n-th stage to the structure of A222180.

Original entry on oeis.org

0, 1, 5, 10, 10, 10, 20, 30, 20, 10, 20, 40, 40, 30, 50, 70, 40, 10, 20, 40, 60, 80
Offset: 0

Views

Author

Omar E. Pol, Mar 15 2013

Keywords

Comments

Essentially the first differences of A222180.
Also number of P-toothpicks added at n-th stage to the P-toothpick structure of A222180.

Examples

			Apparently this is an irregular triangle:
0;
1;
5;
10,10;
10,20,30,20;
10,20,40,40,30,50,70,40;
10,20,40,60,80,...

Links

Crossrefs

Cf. A139250, A139251, A147582, A151724, A152968, A161645, A182633, A182841, A222173, A222177, A222180.

Formula

a(n) = 10*A222173(n-2), n >= 3. - Omar E. Pol, Nov 24 2013

A233971 Number of toothpicks added at n-th stage to the structure of A233970.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 24, 22, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 22, 10, 18, 28, 38, 28, 46, 56, 54, 8, 14, 22, 26, 22, 42, 56, 62, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10
Offset: 0

Views

Author

Omar E. Pol, Dec 18 2013

Keywords

Comments

Essentially the first differences of A233970.
First differs from A170905 at a(24).
First differs from both A233765 and A233781 at a(25).

Examples

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,24,22,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,22,10,18,28,38,28,46,56,54,8,14,22,26,22,42,56,62,16,30,46,58,32,62,64;
Right border gives A000079.

Crossrefs

Cf. A000079, A011782, A139251, A161207, A161831, A170905, A182633, A182635, A182841, A233761, A233765, A233781, A233970, A233972.

A182837 Number of toothpick added at n-th stage to the toothpick structure of A182836.

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 12, 12, 20, 20, 16, 20
Offset: 0

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

First differences of A182836.

Links

Crossrefs

Cf. A139250, A139251, A152980, A182633, A182635, A182836, A182841.

A183005 Number of toothpicks added at n-th stage to the toothpick structure of A183004.

Original entry on oeis.org

0, 1, 4, 6, 8, 8, 16, 22, 16, 8, 16, 24, 24, 32, 56, 62, 32, 8, 16, 24, 24, 32, 56, 64, 40, 32, 56, 72, 80, 120, 176, 158, 64, 8, 16, 24, 24, 32, 56, 64, 40, 32, 56, 72, 80, 120, 176
Offset: 0

Views

Author

Omar E. Pol, Mar 27 2011

Keywords

Comments

Essentially the first differences of A183004.

Examples

			If written as a triangle begins:
  0,
  1,
  4,
  6,8,
  8,16,22,16,
  8,16,24,24,32,56,62,32,
  8,16,24,24,32,56,64,40,32,56,72,80,120,176,158,64,

Links

Crossrefs

Cf. A139250, A139251, A182841, A183004.
Showing 1-8 of 8 results.

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