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 1-7 of 7 results.

A160723 First differences of A160722.

Original entry on oeis.org

1, 4, 4, 10, 4, 10, 10, 22, 4, 10, 10, 22, 10, 22, 22, 46, 4, 10, 10, 22, 10, 22, 22, 46, 10, 22, 22, 46, 22, 46, 46, 94, 4, 10, 10, 22, 10, 22, 22, 46, 10, 22, 22, 46, 22, 46, 46, 94, 10, 22, 22, 46, 22, 46, 46, 94, 22, 46, 46, 94, 46, 94, 94, 190
Offset: 1

Views

Author

Omar E. Pol, May 25 2009

Keywords

Links

Crossrefs

Cf. A139250, A139251, A160720, A160721, A160722.

Formula

a(n) = 3*A001316(n-1) - 2

Extensions

Formula and more terms from Max Alekseyev, Sep 08 2011

A160721 First differences of A160720.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 28, 4, 12, 12, 28, 12, 28, 28, 60, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 12, 28, 28, 60, 28, 60, 60, 124, 28, 60, 60, 124, 60, 124, 124, 252, 4, 12, 12, 28, 12, 28, 28
Offset: 1

Views

Author

Omar E. Pol, May 25 2009, May 29 2009

Keywords

Comments

This sequence is related to the Sierpinski triangle and to Gould's sequence A001316. - Omar E. Pol, Jul 23 2009
When written as a irregular triangle in which row lengths are A011782 it appears that right border gives A173033. - Omar E. Pol, Mar 20 2013

Examples

			From _Omar E. Pol_, Mar 20 2013 (Start):
Triangle begins:
1;
4;
4,12;
4,12,12,28;
4,12,12,28,12,28,28,60;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124,12,28,28,60,28,60,60,124,28,60,60,124,60,124,124,252;
(End)

Links

Crossrefs

Cf. A000120, A001316, A038573, A139250, A139251, A160720, A160722, A160723.

Formula

a(1)=1. Observation: It appears that a(n) = 4*A038573(n-1), n>1. [From Omar E. Pol, Jul 23 2009]. This formula is correct! - N. J. A. Sloane, Jan 23 2016

Extensions

More terms from R. J. Mathar, Jul 14 2009

A327330 "Concave pentagon" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 63, 75, 89, 111, 133, 149, 153, 161, 171, 183, 197, 219, 241, 261, 275, 299, 327, 361, 403, 463, 511, 547, 551, 559, 569, 581, 595, 617, 639, 659, 673, 697, 725, 759, 801, 861, 909, 949, 967, 995, 1029, 1075, 1125, 1183, 1233, 1281, 1321, 1389, 1465, 1549, 1657
Offset: 0

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks (A160164) and V-toothpicks (A161206).
The surprising fact is that after 2^k stages the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two quadrilaterals (Q1 and Q2), both with their largest sides in vertical position, as shown below:
.
* *
* * * *
* * * *
* * *
* Q1 * Q2 *
* * * *
* * * *
* * * *
* * * *
* * E * *
* * * *
* * * *
** **
* * * * * * * * * *
.
Note that for n >> 1 both quadrilaterals look like right triangles.
Every polygon has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.
If n is odd then we add I-toothpicks in vertical position (see the example).
a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
A327331 (the first differences) gives the number of elements added at the n-th stage.
2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
The structure is almost identical to the structure of A327332, but a little larger at the upper edge.
The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
For another version, very similar, starting with a V-toothpick, see A327332, which it appears that shares infinitely many terms with this sequence.

Examples

			Illustration of initial terms:
.
               |      /|\     |/|\|
               |       |      | | |
                      / \     |/ \|
                              |   |
n   :  0       1       2        3
a(n):  0       1       3        7
After three generations there are five I-toothpicks and two V-toothpicks in the structure, so a(3) = 5 + 2 = 7 (note that in total there are 2*a(3) = 2*7 = 14 single toothpicks of length 1).

Links

Crossrefs

First differs from A231348 at a(11).
Cf. A047999, A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327331 (first differences), A327332 (another version).
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Formula

Conjecture: a(2^k) = A327332(2^k), k >= 0.

A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 33, 41, 45, 51, 63, 75, 85, 101, 133, 149, 153, 159, 171, 183, 193, 209, 241, 261, 273, 291, 327, 363, 389, 431, 515, 547, 551, 557, 569, 581, 591, 607, 639, 659, 671, 689, 725, 761, 787, 829, 913, 953, 969, 993, 1041, 1085, 1109, 1149, 1229, 1277, 1309, 1357, 1453, 1549, 1613
Offset: 0

Views

Author

Omar E. Pol, Sep 01 2019

Keywords

Comments

Another version and very similar to A327330.
This arises from a hybrid cellular automaton on a triangular grid formed of V-toothpicks (A161206) and I-toothpicks (A160164).
After 2^k stages, the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two right triangles (R1 and R2) both with their hypotenuses in vertical position, as shown below:
.
* *
* * * *
* * * *
* * *
* R1 * * R2 *
* * * *
* * * *
* * * *
* * E * *
* * * *
* * * *
** **
* * * * * * * * * *
.
Every triangle has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an V-toothpick, formed of two single toothpicks, with its central vertice directed up, like a gable roof, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a I-toothpick formed of two single toothpicks in vertical position.
If n is odd then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof (see the example).
a(n) gives the total number of V-toothpicks and I-toothpicks in the structure after the n-th stage.
A327333 (the first differences) gives the number of elements added at the n-th stage.
2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
The structure is almost identical to the structure of A327330, but a little smaller.
The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
It appears that A327330 shares infinitely many terms with this sequence.

Examples

			Illustration of initial terms:
.
.             /\     |/\|
.                    |  |
.
n:     0       1       2
a(n):  0       1       3
After two generations there are only one V-toothpick and two I-toothpicks in the structure, so a(2) = 1 + 2 = 3 (note that in total there are 2*a(2)= 2*3 = 6 single toothpicks of length 1).

Links

Crossrefs

Cf. A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327333 (first differences), A327330 (another version).
For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.

Formula

Conjecture: a(2^k) = A327330(2^k), k >= 0.

A160744 a(n) = A151566(n)*3.

Original entry on oeis.org

0, 3, 6, 12, 18, 24, 30, 42, 54, 60, 66, 78, 90, 102, 114, 138, 162, 168, 174, 186, 198, 210, 222, 246, 270, 282, 294, 318, 342, 366, 390, 438, 486, 492, 498, 510, 522, 534, 546, 570, 594, 606, 618, 642, 666, 690, 714, 762, 810, 822, 834, 858, 882, 906, 930
Offset: 0

Views

Author

Omar E. Pol, May 25 2009

Keywords

Links

Crossrefs

Cf. A139250, A151566, A160720, A160722, A160736, A160738, A160740, A160742, A160746.

Extensions

More terms from R. J. Mathar, Jul 28 2009

A160742 a(n) = A151566(n)*2.

Original entry on oeis.org

0, 2, 4, 8, 12, 16, 20, 28, 36, 40, 44, 52, 60, 68, 76, 92, 108, 112, 116, 124, 132, 140, 148, 164, 180, 188, 196, 212, 228, 244, 260, 292, 324, 328, 332, 340, 348, 356, 364, 380, 396, 404, 412, 428, 444, 460, 476, 508, 540, 548, 556, 572, 588, 604, 620, 652, 684
Offset: 0

Views

Author

Omar E. Pol, May 25 2009

Keywords

Links

Crossrefs

Cf. A139250, A151566, A160720, A160722, A160736, A160738, A160740, A160744, A160746.

Extensions

More terms from Jinyuan Wang, Mar 14 2020

A160746 a(n) = A151566(n)*4.

Original entry on oeis.org

0, 4, 8, 16, 24, 32, 40, 56, 72, 80, 88, 104, 120, 136, 152, 184, 216, 224, 232, 248, 264, 280, 296, 328, 360, 376, 392, 424, 456, 488, 520, 584, 648, 656, 664, 680, 696, 712, 728, 760, 792, 808, 824, 856, 888, 920, 952, 1016, 1080, 1096, 1112, 1144, 1176, 1208, 1240
Offset: 0

Views

Author

Omar E. Pol, May 25 2009

Keywords

Links

Crossrefs

Cf. A139250, A151566, A160720, A160722, A160736, A160738, A160740, A160742, A160744.

Extensions

More terms from N. J. A. Sloane, May 25 2009
Showing 1-7 of 7 results.

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