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Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1.
2

%I #26 Jul 12 2017 05:29:58

%S 1,7,7,29,7,49,29,103,7,49,49,203,29,203,103,373,7,49,49,203,49,343,

%T 203,721,29,203,203,841,103,721,373,1407,7,49,49,203,49,343,203,721,

%U 49,343,343,1421,203,1421,721,2611,29,203,203,841,203,1421,841,2987,103,721,721,2987,373,2611,1407,5277

%N Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1.

%C This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

%C This is the odd-rule cellular automaton defined by OddRule 575 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

%C Run Length Transform of A246038.

%C The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

%H Alois P. Heinz, <a href="/A246039/b246039.txt">Table of n, a(n) for n = 0..8192</a>

%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796, 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.

%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249, 2015.

%H N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168, 2015

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%e Here is the neighborhood:

%e [X, X, X]

%e [0, X, 0]

%e [X, X, X]

%e which contains a(1) = 7 ON cells.

%p C:=f->subs({x=1, y=1}, f);

%p # Find number of ON cells in CA for generations 0 thru M defined by rule

%p # that cell is ON iff number of ON cells in nbd at time n-1 was odd

%p # where nbd is defined by a polynomial or Laurent series f(x, y).

%p OddCA:=proc(f, M) global C; local n, a, i, f2, p;

%p f2:=simplify(expand(f)) mod 2;

%p a:=[]; p:=1;

%p for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:

%p lprint([seq(a[i], i=1..nops(a))]);

%p end;

%p f:=(1/x+1+x)*(1/y+y)+1 mod 2;

%p OddCA(f, 70);

%t (* f = A246038 *) f[0]=1; f[1]=7; f[2]=29; f[3]=103; f[4]=373; f[n_] := f[n] = 8 f[n-4] + 8 f[n-3] + 3 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* _Jean-François Alcover_, Jul 12 2017 *)

%Y Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035. A246037.

%Y Cf. A246038.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Aug 21 2014