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%I #23 Feb 24 2021 02:48:19
%S 0,1,1,2,2,4,5,4,2,4,6,6,8,14,15,8,2,4,6,6,8,14,16,10,8,14,18,20,30,
%T 44,39,16,2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,
%U 42,28,30,46,56,70,104,128,95
%N Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.
%C It appears that both a(2) and a(2^k - 1) are odd numbers, for k >= 2. Other terms are even numbers.
%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(1)=0. a(n) = A187213(n)/2, for n >= 2.
%F It appears that a(2^k - 1) = A099035(k-1), for k >= 2.
%e At stage 1 we start in the first quadrant from a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1). There are no gulls in the structure, so a(1) = 0.
%e At stage 2 we place a gull (or G-toothpick) with its midpoint at (1,1) and its endpoints at (2,0) and (2,2), so a(2) = 1. There is only one exposed midpoint at (2,2).
%e At stage 3 we place a gull with its midpoint at (2,2), so a(3) = 1. There are two exposed endpoints.
%e At stage 4 we place two gulls, so a(4) = 2. There are two exposed endpoints.
%e At stage 5 we place two gulls, so a(5) = 2. There are four exposed endpoints.
%e And so on.
%e If written as a triangle begins:
%e 0,
%e 1,
%e 1,2,
%e 2,4,5,4,
%e 2,4,6,6,8,14,15,8,
%e 2,4,6,6,8,14,16,10,8,14,18,20,30,44,39,16,
%e 2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,42,28,...
%e It appears that rows converge to A151688.
%p read("transforms3") ;
%p L := BFILETOLIST("b187212.txt") ;
%p A187213 := DIFF(L) ;
%p seq( op(n,A187213)/2,n=2..nops(A187213)) ; # _R. J. Mathar_, Mar 30 2011
%Y Cf. A099035, A151550, A151688, A187210, A187211, A187212, A187213, A187220.
%K nonn
%O 1,4
%A _Omar E. Pol_, Mar 22 2011, Apr 06 2011
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