A327333 -id:A327333 - cp's OEIS frontend

A123932 a(0) = 1, a(n) = 4 for n > 0.

1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Philippe Deléham, Nov 28 2006

Continued fraction for sqrt(5)-1.
a(n) = number of permutations of length n+3 having only one ascent such that the first element of the permutation is 3. - Ran Pan, Apr 20 2015
Also, decimal expansion of 13/90. - Bruno Berselli, Apr 24 2015
Column 1 of A327331 and of A327333. - Omar E. Pol, Nov 25 2019

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

Essentially the same as A113311, A040002 and A010709.

[4^n mod 12: n in [0..40]]; // Vincenzo Librandi, Apr 23 2015

ContinuedFraction[Sqrt[5] - 1, 120] (* Michael De Vlieger, Apr 20 2015 *)

makelist(if n=0 then 1 else 4, n, 0, 100); /* Bruno Berselli, Apr 24 2015 */

a(n)=(n>=0)+3*(n>0) \\ Jaume Oliver Lafont, Mar 26 2009

[power_mod(4, n, 12) for n in range(0, 84)] # Zerinvary Lajos, Nov 25 2009

G.f.: (1 + 3*x) / (1 - x).
a(n) = 4 - 3*0^n .
a(n) = 4^n mod 12. - Zerinvary Lajos, Nov 25 2009
E.g.f.: 4*exp(x) - 3. - Elmo R. Oliveira, Aug 06 2024

A323647 Number of elements added at n-th stage to the toothpick structure of A323646.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 6, 12, 14, 12, 6, 12, 14, 16, 18, 32, 34, 20, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 40, 18, 32, 38, 44, 62, 92, 86, 60, 62, 96, 114, 144, 210, 260, 194, 68, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92

Offset: 1

Omar E. Pol, Mar 07 2019

The "word" of this cellular automaton is "ab", but note that this triangle has an unusual structure: an additional row of length 2. For more information about the word of cellular automata see A296612.
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,a,b;
...
Row lengths give 2 together with the terms of A011782 multiplied by 2, also 2 togheter with the column 2 of A296612.
Columns "a" contain numbers of toothpicks of length 2.
Columns "b" contain numbers of D-toothpicks of length 2*sqrt(2). See the example.

			Triangle begins:
1, 2;
2, 4;
6, 6;
6,12,14,12;
6,12,14,16,18,32,34,20;
6,12,14,16,18,32,34,24,18,32,38,44,62,92,82,36;
6,12,14,16,18,32,34,24,18,32,38,44,62,92,82,40,18,32,38,44,62,92,86,60,62,96, ...

First differences of A323646.
Also, 1 together with A160731.
Column 1 gives A134201.
Cf. A011782, A139251, A296612.
For other hybrid cellular automata, see A194271, A194701, A220501, A289841, A290221, A294021, A294963, A294981, A299771, A323651, A327331, A327333.

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.

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

Omar E. Pol, Sep 01 2019

nonn

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.

			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).

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.

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

cp's OEIS Frontend

A123932 a(0) = 1, a(n) = 4 for n > 0.

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A323647 Number of elements added at n-th stage to the toothpick structure of A323646.

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

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

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