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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.
10

%I #11 Feb 24 2021 02:48:18

%S 0,5,13,27,41,57,85,123,149,165,193,233,277,337,429,527,577,593,621,

%T 661,705,765,857,957,1025,1085,1181,1305,1453,1665,1945,2187,2285,

%U 2301,2329,2369,2413,2473,2565,2665,2733,2793,2889,3013,3161,3373,3653,3897,4013

%N 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 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 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 Conjecture: a(n) = A147614(n)+2*A139250(n). [From _R. J. Mathar_, Jan 22 2010]

%F The above conjecture is true: each toothpick covers exactly two more grid points than the corresponding toothpick in A147614.

%e a(2)=13:

%e .o-o-o-o-o

%e .....|....

%e .....o....

%e .....|....

%e .....o....

%e .....|....

%e .....o....

%e .....|....

%e .o-o-o-o-o

%Y Cf. A139250, A139251, A147614, A147562, A160118, A160120, A160170, A160430.

%K nonn

%O 0,2

%A _Omar E. Pol_, May 13 2009, May 18 2009

%E Definition revised by _N. J. A. Sloane_, Jan 02 2010.

%E Formula verified and more terms from _Nathaniel Johnston_, Nov 13 2010