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A160406
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Toothpick sequence starting at the vertex of an infinite 90-degree wedge.
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32
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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, 102, 108, 118, 128, 140, 160, 186, 202, 204, 206, 210, 214, 218, 224, 234, 242, 246, 252, 262, 272, 284, 304, 330, 346, 350, 356, 366, 376, 388, 408, 434, 452, 464, 484, 512, 542, 584
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OFFSET
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0,3
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COMMENTS
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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.
Number of toothpick in the structure after n rounds.
The toothpick sequence A139250 is the main entry for this sequence. See also A153000. First differences: A160407.
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LINKS
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Table of n, a(n) for n=0..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
Omar Pol, Illustration of initial terms
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FORMULA
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A139250(n) = 2a(n) + 2a(n+1) - 4n - 1 for n > 0. - N. J. A. Sloane, May 25 2009
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
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MAPLE
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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
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MATHEMATICA
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terms = 62;
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);
P = (G + 2 + x(5-x)/(1-x)^2) x/(2(1+x));
CoefficientList[P + O[x]^terms, x] (* Jean-François Alcover, Nov 03 2018, from Maple *)
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CROSSREFS
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Cf. A139250, A139251, A153000, A153006, A152980, A160407, A160408, A160409.
Cf. A170886-A170895.
Sequence in context: A191146 A220850 A151566 * A113293 A080431 A288732
Adjacent sequences: A160403 A160404 A160405 * A160407 A160408 A160409
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KEYWORD
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nonn
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AUTHOR
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Omar E. Pol, May 23 2009
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EXTENSIONS
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More terms from N. J. A. Sloane, May 25 2009
Definition revised by N. J. A. Sloane, Jan 02 2010
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STATUS
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approved
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