A160407 First differences of toothpick numbers A160406.

1, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 6, 10, 8, 2, 2, 4, 4, 4, 6, 10, 8, 4, 6, 10, 10, 12, 20, 26, 16, 2, 2, 4, 4, 4, 6, 10, 8, 4, 6, 10, 10, 12, 20, 26, 16, 4, 6, 10, 10, 12, 20, 26, 18, 12, 20, 28, 30, 42

Offset

1,3

Comments

Number of toothpicks added at n-th stage in the toothpick structure of A160406.

From Omar E. Pol, Mar 15 2020: (Start)

The cellular automaton described in A160406 has word "ab", so the structure of this triangle is as follows:

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;

...

The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...

This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks that are parallel to initial toothpick, and the even-indexed columns (b) contain numbers of the toothpicks that are orthogonal to the initial toothpick.

For further information about the "word" of a cellular automaton see A296612. (End)

Links

Table of n, a(n) for n=1..61.

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

Example

From Omar E. Pol, Jul 18 2009, Mar 15 2020: (Start)
If written as a triangle:
1,1;
2,2;
2,2,4,4;
2,2,4,4,4,6,10,8;
2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16;
2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16,4,6,10,10,12,20,26,18,12,20,28,30,42;...
(End)

Crossrefs

Cf. A011782, A139250, A139251, A153000, A153006, A152980, A160406, A161830, A161831, A296612.

Sequence in context: A051464 A151565 A060632 * A007457 A119802 A237120

Adjacent sequences: A160404 A160405 A160406 * A160408 A160409 A160410

Keyword

nonn

Author

Omar E. Pol, May 23 2009

Extensions

More terms from N. J. A. Sloane, Jul 17 2009

Status

approved