A323647 -id:A323647 - cp's OEIS frontend

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

0, 1, 3, 5, 9, 15, 21, 27, 39, 53, 65, 71, 83, 97, 113, 131, 163, 197, 217, 223, 235, 249, 265, 283, 315, 349, 373, 391, 423, 461, 505, 567, 659, 741, 777, 783, 795, 809, 825, 843, 875, 909, 933, 951, 983, 1021, 1065, 1127, 1219, 1301, 1341, 1359, 1391, 1429, 1473, 1535, 1627, 1713, 1773, 1835, 1931

Offset: 0

Omar E. Pol, Mar 07 2019

nonn

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).
For the construction of the sequence the rules are as follows:
On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.
For the next n generations we have that:
At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.
If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.
If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.
If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.
The sequence gives the number of toothpicks and D-toothpicks after n stages.
A323647 (the first differences) gives the number of elements added at the n-th stage.
Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

			After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.
Note that angle between both D-toothpicks is 90 degrees.
.
                      *
                    *   *
                  * * * * *
                *           *
              *               *
.
After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.
.
                      *
                    *   *
                  * * * * *
                *           *
          * * * * *       * * * * *
.

Cf. A139250, A168112, A160730, A296612, A323647.

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

a(n) = 1 + A160730(n-1), n >= 1.

a(n) = 1 + 2*A168112(n-1), n >= 1.

A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 10, 8, 4, 8, 10, 12, 14, 22, 22, 16, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 36, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40, 18, 28, 34, 46, 50, 58, 50, 48, 40, 68, 76, 84, 108, 156, 100, 76, 4, 8, 10, 12, 14, 22, 22, 20, 14, 24, 28, 34, 42, 60, 48, 40

Offset: 1

Omar E. Pol, Sep 01 2019

The word of this cellular automaton is "ab".
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;
...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of I-toothpicks.
Columns "b" contain numbers of V-toothpicks.
For further information about the word of cellular automata see A296612.

			Triangle begins:
1,2;
4,4;
4,8,10,8;
4,8,10,12,14,22,22,16;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,36;
4,8,10,12,14,22,22,20,14,24,28,34,42,60,48,40,18,28,34,46,50,58,50,48,40,68,...

First differences of A327330.
Column 1 gives A123932.
First differs from A231348 at a(11).
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327333 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

A327333 Number of elements added at n-th stage to the toothpick structure of A327332.

Original entry on oeis.org

1, 2, 4, 4, 4, 6, 12, 8, 4, 6, 12, 12, 10, 16, 32, 16, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 32, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 40, 16, 24, 48, 44, 24, 40, 80, 48, 32, 48, 96, 96, 64, 104, 208, 64, 4, 6, 12, 12, 10, 16, 32, 20, 12, 18, 36, 36, 26, 42, 84, 40

Offset: 1

Omar E. Pol, Sep 01 2019

The word of this cellular automaton is "ab".
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;
...
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
Columns "a" contain numbers of V-toothpicks. Columns "b" contain numbers of I-toothpicks. See the example.
For further information about the word of cellular automata see A296612.

			Triangle begins:
1,2;
4,4;
4,6,12,8;
4,6,12,12,10,16,32,16;
4,6,12,12,10,16,32,20,12,18,36,36,26,42,84,32;
4,6,12,12,10,16,32,20,12,18,36,36,26,42,84,40,16,24,48,44,24,40,80,48,32,48,...
It appears that right border gives the even powers of 2.

First differences of A327332.
Column 1 gives A123932.
Cf. A011782, A139250, A139251, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649, A327331 (similar triangle).
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323647, A323651.

A134201 Number of rigid hypergroups of order n.

Original entry on oeis.org

1, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Roman Bayon (roman.bayon(AT)gmail.com), Oct 14 2007

a(n) is also the number of I-toothpicks added to the structure of the cellular automaton of A323646 when starts its n-th cycle. Column 1 of triangle A323647. - Omar E. Pol, Nov 25 2019

Also decimal expansion of 19/15. - Stefano Spezia, Mar 23 2022

R. Bayon and N. Lygeros, Hyperstructures and Automorphism Groups, submitted.
F. Marty, Sur une généralisation de la notion de groupe. In Proc. 8th Congr. des Mathématiciens Scandinaves, Stockholm, pp. 45-49, 1934.
Th. Vougiouklis, The fundamental relation in hyperrings: The general hyperfield, Fourth Int. Congress Algebraic Hyperstructures and Appl. (AHA), 1991, pp. 203-211.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A108089, A132590, A134202, A323646, A323647.

Array[If[# <= 2, #, 6] &, 105] (* Michael De Vlieger, Dec 01 2019 *)

a(1) = 1, a(2) = 2, a(n) = 6 for n > 2.
G.f.: x*(1 + x + 4*x^2)/(1 - x). - Stefano Spezia, Mar 23 2022
E.g.f.: 6*exp(x) - 6 - 5*x - 2*x^2. - Elmo R. Oliveira, Aug 09 2024

cp's OEIS Frontend

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

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A327331 Number of elements added at n-th stage to the toothpick structure of A327330.

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A327333 Number of elements added at n-th stage to the toothpick structure of A327332.

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A134201 Number of rigid hypergroups of order n.

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