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 1-6 of 6 results.

A233781 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the D-toothpick "wide" triangle of A233780.

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, 18, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 18, 10, 18, 30, 38, 26, 42, 52, 34, 8, 14, 22, 26, 22, 42, 56, 50, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2013

Keywords

Comments

Essentially the first differences of A233780.
First differs from A170905 at a(24).
First differs from A233971 at a(25).
First differs from A233765 at a(44).

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;
2,4,6,10,10,10,18,24,18,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,18,10,18,30,38,26,42,52,34,8,14,22,26,22,42,56,50,16,30,46,58,32,62,64;
Right border gives A000079.

Links

Crossrefs

Cf. A000079, A011782, A139251, A170905, A220501, A220521, A231349, A233765, A233780, A233782, A233971.

A231348 Number of triangles after n-th stage in a cellular automaton based in isosceles triangles of two sizes (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 65, 81, 91, 111, 133, 149, 153, 161, 173, 189, 201, 225, 253, 285, 295, 315, 343, 383, 405, 449, 495, 527, 531, 539, 551, 567, 579, 603, 631, 663, 675, 699, 731, 779, 807, 863, 923, 987, 997, 1017, 1045, 1085, 1113, 1169, 1233, 1313, 1335, 1379, 1439, 1527, 1573, 1665, 1759, 1823
Offset: 0

Views

Author

Omar E. Pol, Dec 15 2013

Keywords

Comments

On the semi-infinite square grid the structure of this C.A. contains "black" triangles and "gray" triangles (see the Links section). Both types of triangles have two sides of length 5^(1/2). Every black triangle has a base of length 2 hence its height is 2 and its area is 2. Every gray triangle has a base of length 2^(1/2) hence its height is 3/(2^(1/2)) and its area is 3/2. Both types of triangles are arranged in the same way as the triangles of Sierpinski gasket (see A047999 and A006046). The black triangles are arranged in vertical direction. On the other hand the gray triangles are arranged in diagonal direction in the holes of the structure formed by the black triangles. Note that the vertices of all triangles coincide with the grid points.
The sequence gives the total number of triangles (black and gray) in the structure after n-th stage. A231349 (the first differences) gives the number of triangles added at n-th stage.
For a more complex structure see A233780.

Examples

			We start at stage 0 with no triangles, so a(0) = 0.
At stage 1 we add a black triangle, so a(1) = 1.
At stage 2 we add two black triangles, so a(2) = 1+2 = 3.
At stage 3 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(3) = 3+2+2 = 7.
At stage 4 we add four black triangles, so a(4) = 7+4 = 11.
At stage 5 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(5) = 11+2+2 = 15.
At stage 6 we add four black triangles and four gray triangles, so a(6) = 15+4+4 = 23.
At stage 7 we add four black triangles and six gray triangles, so a(7) = 23+4+6 = 33.
At stage 8 we add eight black triangles, so a(8) = 33+8 = 41.
And so on.
Note that always we add both black triangles and gray triangles except if n is a power of 2. In this case at stage 2^k we add only 2^k black triangles, for k >= 0.

Links

Crossrefs

Cf. A047999, A006046, A139250, A151566, A160406, A173530, A182634, A194444, A220524, A220478, A230981, A231349, A233780.

A233970 Toothpick sequence on hexagonal net starting from the vertex of a 60-degree wedge (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 205, 213, 227, 249, 275, 291, 321, 353, 355, 359, 365, 375, 385, 395, 413, 437, 459, 469, 487, 515, 553, 581, 627, 683, 737, 745, 759, 781, 807
Offset: 0

Views

Author

Omar E. Pol, Dec 18 2013

Keywords

Comments

Toothpicks are connected by their endpoints. The toothpicks placed in north direction are prohibited. The sequence gives the number of toothpicks after n-th stage in the structure. A233971 (the first differences) give the number of toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from both A233764 and A233780 at a(25).

Links

Crossrefs

Cf. A139250, A161206, A161830, A169780, A182632, A182634, A182634, A182840, A233760, A233764, A233780, A233971, A233972.

A233760 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the third kind, second version.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 53, 67, 79, 93, 109, 111, 115, 121, 133, 147, 167, 189, 215, 237, 249, 267, 299, 337, 365, 395, 427, 429, 433, 439, 451, 465, 485, 507, 533, 555, 575, 605, 649, 711, 763, 809, 863, 901, 913, 931, 967, 1013
Offset: 0

Views

Author

Omar E. Pol, Dec 16 2013

Keywords

Comments

The structure is essentially the same as A220520 but here the borders do not contain toothpicks with exposed endpoints except the initial toothpick. The structure is one of the oblique wedges of several D-toothpick structures. For more information see A220500. The first differences (A233761) give the number of toothpicks or D-toothpicks added at n-th stage.

Links

Crossrefs

Cf. A139250, A220500, A220520, A233761, A233762, A233780.

A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 507, 545, 575, 625, 679, 713, 721, 735, 757, 783
Offset: 0

Views

Author

Omar E. Pol, Dec 16 2013

Keywords

Comments

The D-toothpicks placed in northwest or northeast direction both are prohibited. Note that due this rule there are substructures with broken symmetry, for instance a(44) = 507, not 509. For another version without broken symmetry see A233780.
A233765 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233780 at a(44).

Links

Crossrefs

Cf. A139250, A169780, A220500, A220520, A231348, A233765, A233780, A233970.

A233782 a(n) = A233781(n+1)/2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 12, 9, 4, 7, 11, 13, 8, 15, 16, 1, 2, 3, 5, 5, 5, 9, 12, 9, 5, 9, 15, 19, 13, 21, 26, 17, 4, 7, 11, 13, 11, 21, 28, 25, 8, 15, 23, 29, 16, 31, 32, 1, 2, 3, 5, 5, 5, 9, 12, 9, 5, 9, 15, 19
Offset: 1

Views

Author

Omar E. Pol, Dec 17 2013

Keywords

Comments

First 22 terms coincide with a(2)-a(23) of A169778.

Examples

			Written as an irregular triangle in which the row lengths is A000079 the sequence begins:
1;
1,2;
1,2,3,4;
1,2,3,5,5,4,7,8;
1,2,3,5,5,5,9,12,9,4,7,11,13,8,15,16;
1,2,3,5,5,5,9,12,9,5,9,15,19,13,21,26,17,4,7,11,13,11,21,28,25,8,15,23,29,16,31,32;

Links

Crossrefs

Right border gives A000079.
Cf. A139251, A169778, A220501, A220521, A231351, A233781, A233780.
Showing 1-6 of 6 results.

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