David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://neilsloane.com/docA000695/tootha000695_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.]

David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://neilsloane.com/doc/tooth.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.]

N. J. A. Sloane, <a href="http://neilsloane.com/docwiki/toothlist.htmlCatalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.attneilsloane.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>

N. J. A. Sloane, <a href="http://www.research.attneilsloane.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

_N. J. A. Sloane (njas(AT)research.att.com), _, May 24 2009

Another version of the toothpick sequence A139250 (see Comments for definition).

0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 79, 91, 111, 139, 155, 159, 167, 179, 191, 203, 223, 251, 271, 283, 303, 331, 359, 387, 431, 491, 523, 527, 535, 547, 559, 571, 591, 619, 639, 651, 671, 699, 727, 755, 799, 859, 895, 907, 927, 955, 983, 1011, 1055, 1115, 1159, 1187

0,3

The idea is to build a version of A139250 from four copies of the triangle in A151566 (each rotated from the previous one by 90 degrees). The result doesn't quite match A139250, however.

The toothpicks here have length 2, and are placed on the square grid Z X Z.

Place a vertical toothpick centered at (0,0) and extend it downwards to form an infinite triangle using the rule for leftist trees in A151566.

Place another vertical toothpick centered at (0,0) and extend it upwards to form an infinite triangle using the rule for leftist trees in A151566.

Place a horizontal toothpick centered at (1,0) and extend it leftwards to form an infinite triangle using the rule for leftist trees in A151566, then remove the toothpick centered at (1,0).

Place another horizontal toothpick centered at (-1,0) and extend it rightwards to form an infinite triangle using the rule for leftist trees in A151566, then remove the toothpick centered at (-1,0).

Finally, coalesce any toothpicks that have been superimposed. The result starts like A139250, but after 12 generations has fewer toothpicks.

The sequence gives the number of toothpicks in the n-th generation.

David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>

N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

a(n) = 2*P(n) + 2*P(n+1) -4*n - 1, where P() = A151566().

nonn

N. J. A. Sloane (njas(AT)research.att.com), May 24 2009

approved