cp's OEIS Frontend

A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).

%I A246037 #31 Jul 12 2017 05:31:42
%S A246037 1,6,6,20,6,36,20,88,6,36,36,120,20,120,88,336,6,36,36,120,36,216,120,
%T A246037 528,20,120,120,400,88,528,336,1376,6,36,36,120,36,216,120,528,36,216,
%U A246037 216,720,120,720,528,2016,20,120,120,400,120,720,400,1760,88,528,528,1760,336,2016,1376,5440
%N A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).
%C A246037 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 A246037 This is the odd-rule cellular automaton defined by OddRule 077 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%C A246037 Run Length Transform of A246036.
%C A246037 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 A246037 Alois P. Heinz, <a href="/A246037/b246037.txt">Table of n, a(n) for n = 0..8192</a>
%H A246037 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 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.
%H A246037 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 [math.CO], 2015.
%H A246037 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 A246037 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 [math.CO], 2015.
%H A246037 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A246037 Here is the neighborhood:
%e A246037 [X, X, X]
%e A246037 [0, 0, 0]
%e A246037 [X, X, X]
%e A246037 which contains a(1) = 6 ON cells.
%p A246037 C:=f->subs({x=1, y=1}, f);
%p A246037 # Find number of ON cells in CA for generations 0 thru M defined by rule
%p A246037 # that cell is ON iff number of ON cells in nbd at time n-1 was odd
%p A246037 # where nbd is defined by a polynomial or Laurent series f(x, y).
%p A246037 OddCA:=proc(f, M) global C; local n, a, i, f2, p;
%p A246037 f2:=simplify(expand(f)) mod 2;
%p A246037 a:=[]; p:=1;
%p A246037 for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
%p A246037 lprint([seq(a[i], i=1..nops(a))]);
%p A246037 end;
%p A246037 f:=(1/x+1+x)*(1/y+y);
%p A246037 OddCA(f, 70);
%t A246037 (* f = A246036 *) f[0] = 1; f[n_] := (4^(n+1)-(-2)^n)/3; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* _Jean-François Alcover_, Jul 12 2017 *)
%Y A246037 Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035.
%Y A246037 Cf. A246036.
%K A246037 nonn
%O A246037 0,2
%A A246037 _N. J. A. Sloane_, Aug 21 2014