

A152998


Toothpick sequence on the semiinfinite square grid.


14



0, 1, 3, 5, 7, 11, 17, 21, 23, 27, 33, 39, 47, 61, 77, 85, 87, 91, 97, 103, 111, 125, 141, 151, 159, 173, 191, 211, 241, 285, 325, 341, 343, 347, 353, 359, 367, 381, 397, 407, 415, 429, 447, 467, 497, 541, 581, 599, 607, 621, 639
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OFFSET

0,3


COMMENTS

Contribution from Omar E. Pol, Oct 01 2011 (Start):
On the semiinfinite 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 nth stage.
For more information see A139250. (End)


LINKS

Table of n, a(n) for n=0..50.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences


FORMULA

a(n) = (A139250(n+1)1)/2.
Contribution from Omar E. Pol, Oct 01 2011 (Start):
a(n) = A139250(n+1)  A153003(n) + A153000(n1)  1, if n >= 1.
a(n) = A153003(n)  A153000(n1), if n >= 1.
a(n) = 2*A153000(n1) + 1, if n >= 1.
(End)
a(n) = (A187220(n+2)  3)/4.  Omar E. Pol, Feb 18 2013


CROSSREFS

Cf. A139250, A139251, A152968.
Cf. A153000, A152978.
Sequence in context: A235476 A045397 A211409 * A222176 A108539 A170886
Adjacent sequences: A152995 A152996 A152997 * A152999 A153000 A153001


KEYWORD

nonn


AUTHOR

Omar E. Pol, Dec 19 2008, Dec 23 2008, Jan 02 2008


STATUS

approved



