cp's OEIS Frontend

A246039 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1.

%I A246039 #26 Jul 12 2017 05:29:58
%S A246039 1,7,7,29,7,49,29,103,7,49,49,203,29,203,103,373,7,49,49,203,49,343,
%T A246039 203,721,29,203,203,841,103,721,373,1407,7,49,49,203,49,343,203,721,
%U A246039 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 A246039 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1.
%C A246039 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 A246039 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 A246039 Run Length Transform of A246038.
%C A246039 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 A246039 Alois P. Heinz, <a href="/A246039/b246039.txt">Table of n, a(n) for n = 0..8192</a>
%H A246039 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 A246039 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 A246039 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 A246039 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 A246039 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A246039 Here is the neighborhood:
%e A246039 [X, X, X]
%e A246039 [0, X, 0]
%e A246039 [X, X, X]
%e A246039 which contains a(1) = 7 ON cells.
%p A246039 C:=f->subs({x=1, y=1}, f);
%p A246039 # Find number of ON cells in CA for generations 0 thru M defined by rule
%p A246039 # that cell is ON iff number of ON cells in nbd at time n-1 was odd
%p A246039 # where nbd is defined by a polynomial or Laurent series f(x, y).
%p A246039 OddCA:=proc(f, M) global C; local n, a, i, f2, p;
%p A246039 f2:=simplify(expand(f)) mod 2;
%p A246039 a:=[]; p:=1;
%p A246039 for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
%p A246039 lprint([seq(a[i], i=1..nops(a))]);
%p A246039 end;
%p A246039 f:=(1/x+1+x)*(1/y+y)+1 mod 2;
%p A246039 OddCA(f, 70);
%t A246039 (* 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 A246039 Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035. A246037.
%Y A246039 Cf. A246038.
%K A246039 nonn
%O A246039 0,2
%A A246039 _N. J. A. Sloane_, Aug 21 2014