%N Toothpick sequence in the first quadrant.
%C Contribution from Omar E. Pol, Nov 29 2009: (Start)
%C At stage 0, we start from a horizontal half toothpick at [(0,1),(1,1)]. This half toothpick represents one of the two components of the second toothpick placed in the toothpick structure of A139250. Consider only the toothpicks of length 2, so a(0) = 0.
%C At stage 1 we place an orthogonal toothpick of length 2 centered at the end, so a(1) = 1.
%C In each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
%C The sequence gives the number of toothpicks after n stages. Note that this sequence contains even numbers and odd numbers, the same as A152978 (the first differences) which gives the number of toothpicks added at n-th stage. For more information see A139250. (End)
%D D. Applegate, O. E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://neilsloane.com/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, which is also available at <a href="http://arxiv.org/abs/1004.3036">arXiv:1004.3036v2</a>
%H O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp016.jpg">Illustration of initial terms</a> [From Omar E. Pol, Nov 29 2009]
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F a(n) = (A139250(n+2)-3)/4 = (A152998(n+1)-1)/2.
%F G.f.: (1+x)*(Prod(1+x^(2^k-1)+2*x^(2^k),k=1..oo)-1)/((1-x)*(1+2*x)). - _N. J. A. Sloane_, May 20 2009
%F Contribution from Omar E. Pol, Oct 01 2011 (Start):
%F a(n) = A152998(n+1) + A153003(n+1) - A139250(n+2) + 1.
%F a(n) = A139250(n+2) - A153003(n+1) - 2.
%F a(n) = A153003(n+1) - A152998(n+1).
%F a(n) = (A187220(n+3) - 7)/8. - _Omar E. Pol_, Feb 16 2013
%p G := (1+x)*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..20)-1)/((1-x)*(1+2*x)); - _N. J. A. Sloane_, May 20 2009
%Y Cf. A139250, A139251, A152978, A153006.
%Y Cf. A152998, A160406. [From Omar E. Pol, Nov 29 2009]
%A _Omar E. Pol_, Dec 16 2008, Dec 20 2008, Jan 02 2009