A160406 Toothpick sequence starting at the vertex of an infinite 90-degree wedge.

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

Offset

0,3

Comments

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.

Links

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.]

Omar E. Pol, Illustration of initial terms

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Formula

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

Maple

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

Mathematica

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 *)

Crossrefs

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

Keyword

nonn

Author

Omar E. Pol, May 23 2009

Extensions

More terms from N. J. A. Sloane, May 25 2009

Definition revised by N. J. A. Sloane, Jan 02 2010

Status

approved