login
Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).
4

%I #31 Jul 12 2017 05:31:42

%S 1,6,6,20,6,36,20,88,6,36,36,120,20,120,88,336,6,36,36,120,36,216,120,

%T 528,20,120,120,400,88,528,336,1376,6,36,36,120,36,216,120,528,36,216,

%U 216,720,120,720,528,2016,20,120,120,400,120,720,400,1760,88,528,528,1760,336,2016,1376,5440

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

%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 077 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

%C Run Length Transform of A246036.

%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="/A246037/b246037.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 [math.CO], 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 [math.CO], 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 [math.CO], 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, 0, 0]

%e [X, X, X]

%e which contains a(1) = 6 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);

%p OddCA(f, 70);

%t (* 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 Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035.

%Y Cf. A246036.

%K nonn

%O 0,2

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