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A183004 Toothpick sequence on square grid with toothpicks connected by their endpoints.

%I A183004 #12 Feb 24 2021 02:48:19
%S A183004 0,1,5,11,19,27,43,65,81,89,105,129,153,185,241,303,335,343,359,383,
%T A183004 407,439,495,559,599,631,687,759,839,959,1135,1293,1357,1365,1381,
%U A183004 1405,1429,1461,1517,1581,1621,1653,1709,1781
%N A183004 Toothpick sequence on square grid with toothpicks connected by their endpoints.
%C A183004 Rules:
%C A183004 - If n is odd then each new toothpick must lie in vertical direction.
%C A183004 - If n is even then each new toothpick must lie in horizontal direction.
%C A183004 - Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.
%C A183004 The sequence gives the number of toothpicks after n stages. A183005 (the first differences) gives the number added at the n-th stage.
%C A183004 The structure is very similar to the structure of A139250 but the mechanism for the connection of toothpicks is different.
%H A183004 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 A183004 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 A183004 We start at stage 0 with no toothpicks.
%F A183004 At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis, so a(1)=1. There are two exposed endpoints.
%F A183004 At stage 2, place 4 toothpicks in horizontal position: two new toothpicks touching each exposed endpoint, so a(2)=1+4=5. There are 4 exposed endpoints.
%F A183004 At stage 3, place 6 toothpicks in vertical position, so a(3)=5+6=11.
%F A183004 After 3 stages the toothpick structure has 2 squares and 4 exposed endpoints.
%Y A183004 Cf. A139250, A160164, A182840, A183005.
%K A183004 nonn
%O A183004 0,3
%A A183004 _Omar E. Pol_, Mar 27 2011