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.

A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

Original entry on oeis.org

0, 3, 6, 12, 12, 12, 24, 36, 24, 12, 24, 48, 60, 48, 60, 84, 48, 12, 24, 48, 60, 60, 84, 132, 132, 72, 60, 120, 168, 144, 156, 192, 96, 12, 24, 48, 60, 60, 84, 132, 132, 84, 84, 156, 228, 228, 228
Offset: 0

Views

Author

Omar E. Pol, Dec 07 2010

Keywords

Comments

First differences of A182632.
a(n) is also the number of components added at n-th stage in the toothpick structure formed by V-toothpicks with an initial Y-toothpick, since a V-toothpick has two components and a Y-toothpick has three components (For more information see A161206, A160120, A161644).

Examples

			From _Omar E. Pol_, Feb 08 2013 (Start):
When written as a triangle:
0;
3;
6;
12,12;
12,24,36,24;
12,24,48,60,48,60, 84, 48;
12,24,48,60,60,84,132,132,72,60,120,168,144,156,192,96;
12,24,48,60,60,84,132,132,84,84,156,228,228,228,...
...
It appears that positive terms of the right border are A007283.
(End)

Links

Crossrefs

A139250, A139251, A160120, A160121, A161206, A161207, A161644, A161645, A182632.

Formula

It appears that a(n) = 2*A161645(n) but with a(1)=3.
a(n) = 3*A182635(n). - Omar E. Pol, Feb 09 2013

A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

Original entry on oeis.org

0, 1, 4, 8, 14, 16, 14, 24, 38, 32, 14, 24, 46, 64, 54, 56, 86, 64, 14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128, 14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256
Offset: 0

Views

Author

Omar E. Pol, Dec 09 2010

Keywords

Comments

First differences of A182840.

Examples

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle:
0;
1;
4;
8;
14, 16;
14, 24, 38, 32;
14, 24, 46, 64, 54, 56, 86, 64;
14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128;
14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256;
...
(End)

Links

Crossrefs

Cf. A139250, A139251, A160121, A161207, A182633, A182635, A151724, A182842.

Formula

a(2^k + 1) = 2^(k+2), at least for 0 <= k <= 9. - Omar E. Pol, Nov 01 2014

Extensions

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

A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 11, 8, 4, 8, 16, 24, 12, 20, 25, 16, 4, 8, 16, 24, 28, 36, 42, 44, 20, 24, 40, 64, 32, 44, 53, 32, 4, 8, 16, 24, 28, 36, 44, 52, 42, 48, 60, 100, 68, 84, 83, 84, 28, 24, 44, 72, 84, 104, 116, 132, 54, 56, 92, 144, 72, 92, 109, 64, 4
Offset: 0

Views

Author

Omar E. Pol, Aug 24 2011

Keywords

Comments

Essentially the first differences of A194444.
First differs from A220525 at a(13). - Omar E. Pol, Mar 23 2013

Examples

			Contribution from _Omar E. Pol_, Dec 05 2012 (Start):
Triangle begins:
0;
1;
2;
4,4;
4,8,11,8;
4,8,16,24,12,20,25,16;
4,8,16,24,28,36,42,44,20,24,40,64,32,44,53,32;
(End)

Links

Crossrefs

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.
Cf. A139250, A139251, A172311, A182635, A182839, A194271, A194433, A194435, A194441, A194443, A194444, A212009.

Formula

It appears that a(2^k+1) = 4, if k >= 1.
a(n) = A194435(n)/4. - Omar E. Pol, Mar 23 2013

Extensions

More terms from Omar E. Pol, Mar 23 2013

A182634 Toothpick sequence on hexagonal net starting at the vertex of an infinite 120-degree wedge.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 71, 91, 107, 127, 155, 171, 175, 183, 199, 219, 239, 267, 311, 355, 379, 399, 439, 495, 543, 595, 659, 691, 695, 703, 719, 739, 759, 787, 831, 875, 903, 931, 983, 1059, 1135, 1211
Offset: 0

Views

Author

Omar E. Pol, Dec 08 2010

Keywords

Comments

The sequence gives the number of toothpicks after n stages. A182635 (the first differences) gives the number added at the n-th stage.
The 120-degree wedge defines a conic region which toothpicks (except one end point of the initial toothpick) are not allowed to cross or touch. The wings of the wedge point +-60 degrees away from the pointing direction of the initial toothpick.
Toothpicks are connected by their endpoints, the same as the toothpicks of A182632.
First differs from A139250 at a(11).

Links

Crossrefs

Cf. A139250, A160406, A160120, A161206, A161644, A182632, A182633, A182635.

Formula

a(n) = A182632(n)/3.

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

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.

A225355 a(n) = A221565(n)/4.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 7, 12, 8, 4, 8, 16, 16, 20, 21, 28, 16, 4, 8, 16, 24, 28, 36, 44, 34, 28, 24, 44, 44, 52, 49, 60, 32, 4, 8, 16, 24, 28, 38
Offset: 0

Views

Author

Omar E. Pol, May 13 2013

Keywords

Examples

			When written as a irregular triangle begins:
0;
1;
2;
4,4;
4,7,12,8;
4,8,16,16,20,21,28,16;
4,8,16,24,28,36,44,34,28,24,44,44,52,49,60,32;
4,8,16,24,28,38...

Links

Crossrefs

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.
Cf. A139251, A182635, A221565.
Showing 1-8 of 8 results.

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