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%I #22 Sep 06 2014 16:39:44
%S 0,1,3,3,7,5,9,9,17,11,15,15,23,19,27,27,43,29,33,33,41,37,45,45,61,
%T 49,57,57,73,65,81,81,113,83,87,87,95,91,99,99,115,103,111,111,127,
%U 119,135,135,167,139,147,147,163,155,171,171,203,179,195,195,227,211,243,243,307,245,249,249,257
%N a(0)=0; for n >= 1, a(n) = f(n) - 2*f(floor((n-1)/2)), where f(n) = A006046(n).
%C For n >= 1, a(n) is the number of ON cells in the n-th generation of the 2-D Mitra-Kumar cellular automaton defined as follows. The state of a cell depends on the states of its NW and NE neighbors at the previous generation.
%C An ON cell remains ON iff 0 or 2 of its neighbors were ON, and an OFF cell turns ON iff exactly one of its neighbors was ON.
%D Sugata Mitra and Sujai Kumar. "Fractal replication in time-manipulated one-dimensional cellular automata." Complex Systems, 16.3 (2006): 191-207. See Fig. 16.
%H Reinhard Zumkeller, <a href="/A245550/b245550.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e The first few generations of the automaton are:
%e X 0X0 00X00 000X000 0000X0000
%e X0X 00000 0000000 000000000
%e X000X 0X000X0 00X000X00
%e X0X0X0X 000000000
%e X0000000X
%e The rows are a subset of the rows of Pascal's triangle A007318 read mod 2 (see A006943). The binary weights of these rows are given by A001316, whose partial sums are A006046, and the formula in the definition follows easily from this.
%p f:=proc(n) option remember; # A006046
%p if n <= 1 then n elif n mod 2 = 0 then 3*f(n/2)
%p else 2*f((n-1)/2)+f((n+1)/2); fi; end;
%p g:=n->f(n)-2*f(floor((n-1)/2));
%p [0,seq(g(n),n=1..130)];
%o (Haskell)
%o a245550 n = a245550_list !! n
%o a245550_list = 0 : zipWith (-) (tail a006046_list) (h a006046_list)
%o where h (x:xs) = (2 * x) : (2 * x) : h xs
%o -- _Reinhard Zumkeller_, Jul 29 2014
%Y Cf. A006046, A001316, A006943, A007318.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jul 29 2014
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