A160407 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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;

...

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