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%I #21 Dec 22 2021 11:43:51
%S 0,1,2,4,6,8,10,14,18,20,22,26,30,34,40,50,58,60,62,66,70,74,80,90,98,
%T 102,108,118,128,140,160,186,202,204,206,210,214,218,224,234,242,246,
%U 252,262,272,284,304,330,346,350,356,366,376,388,408,434,452,464,484,512,542,584
%N Toothpick sequence starting at the vertex of an infinite 90-degree wedge.
%C Consider the wedge of the plane defined by points (x,y) with y >= |x|, with the initial toothpick extending from (0,0) to (0,2); then extend by the same rule as for A139250, always staying inside the wedge.
%C Number of toothpick in the structure after n rounds.
%C The toothpick sequence A139250 is the main entry for this sequence. See also A153000. First differences: A160407.
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp406.jpg">Illustration of initial terms</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A139250(n) = 2a(n) + 2a(n+1) - 4n - 1 for n > 0. - _N. J. A. Sloane_, May 25 2009
%F Let G = (x + 2*x^2 + 4*x^2*(1+x)*((Product_{k>=1} (1 + x^(2^k-1) + 2*x^(2^k))) - 1)/(1+2*x))/(1-x) (= g.f. for A139250); then the g.f. for the present sequence is (G + 2 + x*(5-x)/(1-x)^2)*x/(2*(1+x)). - _N. J. A. Sloane_, May 25 2009
%p G := (x + 2*x^2 + 4*x^2*(1+x)*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..20)-1)/(1+2*x))/(1-x); P:=(G + 2 + x*(5-x)/(1-x)^2)*x/(2*(1+x)); series(P,x,200); seriestolist(%); # _N. J. A. Sloane_, May 25 2009
%t terms = 62;
%t G = (x + 2x^2 + 4x^2 (1+x)(Product[1+x^(2^k-1) + 2x^(2^k), {k, 1, Ceiling[ Log[2, terms]]}]-1)/(1+2x))/(1-x);
%t P = (G + 2 + x(5-x)/(1-x)^2) x/(2(1+x));
%t CoefficientList[P + O[x]^terms, x] (* _Jean-François Alcover_, Nov 03 2018, from Maple *)
%Y Cf. A139250, A139251, A153000, A153006, A152980, A160407, A160408, A160409.
%Y Cf. A170886-A170895.
%K nonn
%O 0,3
%A _Omar E. Pol_, May 23 2009
%E More terms from _N. J. A. Sloane_, May 25 2009
%E Definition revised by _N. J. A. Sloane_, Jan 02 2010
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