A152998 Toothpick sequence on the semi-infinite square grid.

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

Offset: 0

Author

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

Keywords

• nonn

Comments

Contribution from Omar E. Pol, Oct 01 2011 (Start):

On the semi-infinite 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 n-th stage.

For more information see A139250. (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..8191
• 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
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A139250, A139251, A152968.

Cf. A153000, A152978.

Formula

a(n) = (A139250(n+1)-1)/2.

From Omar E. Pol, Oct 01 2011: (Start)

a(n) = A139250(n+1) - A153003(n) + A153000(n-1) - 1, if n >= 1.

a(n) = A153003(n) - A153000(n-1), if n >= 1.

a(n) = 2*A153000(n-1) + 1, if n >= 1.

(End)

a(n) = (A187220(n+2) - 3)/4. - Omar E. Pol, Feb 18 2013