A151567 - OEIS

%I #6 Feb 24 2021 02:48:18

%S 0,1,3,7,11,15,23,35,43,47,55,67,79,91,111,139,155,159,167,179,191,

%T 203,223,251,271,283,303,331,359,387,431,491,523,527,535,547,559,571,

%U 591,619,639,651,671,699,727,755,799,859,895,907,927,955,983,1011,1055,1115,1159,1187

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

%C 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.

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

%C 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.

%C 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.

%C 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).

%C 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).

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

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

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_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.]

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

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

%K nonn

%O 0,3

%A _N. J. A. Sloane_, May 24 2009