OFFSET
0,3
COMMENTS
Contribution from Omar E. Pol, Oct 01 2011 (Start):
On the semi-infinite square grid, at stage 0, we start from a vertical half toothpick at [(0,0),(0,1)]. This half toothpick represents one of the two components of the first toothpick placed in the toothpick structure of A139250. Consider only the toothpicks of length 2, so a(0) = 0.
At stage 1, we place an orthogonal toothpick of length 2 centered at the end, so a(1) = 1.
In each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
The sequence gives the number of toothpicks after n stages. A152968 (the first differences) gives the number of toothpicks added to the structure at n-th stage.
For more information see A139250. (End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..8191
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
FORMULA
a(n) = (A139250(n+1)-1)/2.
From Omar E. Pol, Oct 01 2011: (Start)
a(n) = 2*A153000(n-1) + 1, if n >= 1.
(End)
a(n) = (A187220(n+2) - 3)/4. - Omar E. Pol, Feb 18 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 19 2008, Dec 23 2008, Jan 02 2008
STATUS
approved