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.

A220497 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A220496.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 8, 8, 8, 13, 16, 4, 4, 8, 8, 8, 8, 14, 16, 8, 8, 16, 16, 16, 16, 25, 32, 4, 4, 8, 8, 8, 8, 14, 16, 8, 8, 16, 16, 16, 16, 26, 32, 8, 8, 16, 16, 16, 16, 28, 32, 16, 16, 32, 32, 32, 32, 49, 64, 4, 4, 8, 8, 8, 8, 14, 16
Offset: 0

Views

Author

Omar E. Pol, Dec 23 2012

Keywords

Comments

Essentially the first differences of A220496.
First differs from A194443 (and from A220523) at a(12).

Examples

			Written as an irregular triangle begins:
0;
1;
2;
4,4;
4,4,7,8;
4,4,8,8,8,8,13,16;
4,4,8,8,8,8,14,16,8,8,16,16,16,16,25,32;
4,4,8,8,8,8,14,16,8,8,16,16,16,16,26,32,8,8,16,16,16,16,28,32,16,16,32,32,32,32,49,64;

Links

Crossrefs

Cf. A139250, A139251, A194271, A194443, A194700, A194701, A220495, A220496, A220523.

A194442 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the second kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 693, 741, 761, 769, 785, 813, 853, 897, 947
Offset: 0

Views

Author

Omar E. Pol, Aug 29 2011

Keywords

Comments

If n = 2^k, k >= 1, then the structure looks like an isosceles triangle. For the D-toothpick "wide" triangle of the second kind see A194440.
The structure is essentially one of the wedges of several D-toothpick structures. For more information see A194270. The first differences (A194443) give the number of toothpicks or D-toothpicks added at n-th stage. - Omar E. Pol, Mar 28 2013

Links

Crossrefs

Cf. A047999, A139250, A160120, A160406, A161830, A194270, A194277, A194440, A194443, A194444, A194694, A194695, A194697, A220496, A220522.

A220522 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 701, 749, 769, 777
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2012

Keywords

Comments

The structure is essentially one of the oblique wedges of several D-toothpick structures. For more information see A220500. First differs from A194442 at a(47). The first differences (A220523) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A047999, A139250, A194270, A194700, A194440, A194442, A194444, A220496, A220500, A220523.

A220494 Number of toothpicks and D-toothpicks after n-th stage in the structure of the D-toothpick "wide" triangle of the first kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 59, 67, 75, 91, 107, 111, 115, 123, 131, 139, 147, 163, 179, 187, 195, 211, 227, 243, 259, 291, 323, 327, 331, 339, 347, 355, 363, 379, 395, 403, 411, 427, 443, 459, 475, 507, 539, 547, 555, 571, 587, 603
Offset: 0

Views

Author

Omar E. Pol, Dec 23 2012

Keywords

Comments

This cellular automaton uses toothpicks of length 1 and D-toothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
On the semi-infinite square grid we start with no elements, so a(0) = 0. At stage 1, we place a single toothpick in vertical direction at (0,0),(0,1), so a(1) = 1. Note that there is only one exposed toothpick endpoint.
The rules for adding new elements are as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. The endpoints of the toothpicks of the old generation that are perpendiculars to the initial toothpick remain exposed forever. Overlapping is prohibited.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220495 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage.
It appears that the structure has fractal behavior related to powers of 2. It appears that this cellular automaton has a surprising connection with the Sierpinski triangle, but here the structure is more complex.
For a similar version see A220496. For other more complex versions see A194440, A220520. First differs from A194440 (and from A220520) at a(12).

Links

Crossrefs

Cf. A047999, A139250, A194270, A194440, A220495, A220496, A220520.
Showing 1-4 of 4 results.

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