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A160420 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose skeleton is the same network as the toothpick structure of A139250 but with toothpicks of length 4.

%I A160420 #11 Feb 24 2021 02:48:18
%S A160420 0,5,13,27,41,57,85,123,149,165,193,233,277,337,429,527,577,593,621,
%T A160420 661,705,765,857,957,1025,1085,1181,1305,1453,1665,1945,2187,2285,
%U A160420 2301,2329,2369,2413,2473,2565,2665,2733,2793,2889,3013,3161,3373,3653,3897,4013
%N A160420 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose skeleton is the same network as the toothpick structure of A139250 but with toothpicks of length 4.
%C A160420 a(n) is also the number of grid points that are covered after n-th stage by an polyedge as the toothpick structure of A139250, but with toothpicks of length 4.
%H A160420 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 A160420 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 A160420 Conjecture: a(n) = A147614(n)+2*A139250(n). [From _R. J. Mathar_, Jan 22 2010]
%F A160420 The above conjecture is true: each toothpick covers exactly two more grid points than the corresponding toothpick in A147614.
%e A160420 a(2)=13:
%e A160420 .o-o-o-o-o
%e A160420 .....|....
%e A160420 .....o....
%e A160420 .....|....
%e A160420 .....o....
%e A160420 .....|....
%e A160420 .....o....
%e A160420 .....|....
%e A160420 .o-o-o-o-o
%Y A160420 Cf. A139250, A139251, A147614, A147562, A160118, A160120, A160170, A160430.
%K A160420 nonn
%O A160420 0,2
%A A160420 _Omar E. Pol_, May 13 2009, May 18 2009
%E A160420 Definition revised by _N. J. A. Sloane_, Jan 02 2010.
%E A160420 Formula verified and more terms from _Nathaniel Johnston_, Nov 13 2010