A182839 Number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A182838.

0, 1, 2, 4, 4, 4, 6, 10, 8, 4, 6, 12, 16, 14, 14, 22, 16, 4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32, 4, 6, 12, 16, 16, 20, 32, 36, 24

Offset

0,3

Comments

From Omar E. Pol, Feb 06 2023: (Start)

The "word" of this cellular automaton is "ab".

Apart from the initial zero 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;

...

Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).

Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).

An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.

Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.

For further information about the word of cellular automata see A296612.

It appears that the right border of the irregular triangle gives the even powers of 2. (End)

Links

Table of n, a(n) for n=0..41.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to toothpick sequences

Formula

Conjecture: a(n) = (A182841(n+1) + A010673(n))/4, n >= 2. - Omar E. Pol, Feb 10 2023

Example

From Omar E. Pol, Feb 06 2023: (Start)
The nonzero terms can write as an irregular triangle as shown below:
  1, 2;
  4, 4;
  4, 6, 10, 8;
  4, 6, 12, 16, 14, 14, 22, 16;
  4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32;
  ...
(End)

Crossrefs

First differences of A182838.

Cf. A139250, A139251, A161207, A182633, A182635, A182841.

Cf. A000079, A011782, A296612.

Sequence in context: A033827 A008217 A170889 * A043552 A325940 A035545

Adjacent sequences: A182836 A182837 A182838 * A182840 A182841 A182842

Keyword

nonn,tabf,more

Author

Omar E. Pol, Dec 12 2010

Extensions

a(19)-a(41) from Omar E. Pol, Jan 06 2023

Status

approved