cp's OEIS Frontend

A253071 Number of odd terms in f^n, where f = 1/(x*y)+1/x+1/y+y+x/y+x+x*y.

%I A253071 #22 Jul 12 2017 05:29:53
%S A253071 1,7,7,21,7,49,21,95,7,49,49,147,21,147,95,333,7,49,49,147,49,343,147,
%T A253071 665,21,147,147,441,95,665,333,1319,7,49,49,147,49,343,147,665,49,343,
%U A253071 343,1029,147,1029,665,2331,21,147,147,441,147,1029,441,1995,95,665,665,1995,333,2331,1319,4837
%N A253071 Number of odd terms in f^n, where f = 1/(x*y)+1/x+1/y+y+x/y+x+x*y.
%C A253071 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 A253071 This is the odd-rule cellular automaton defined by OddRule 357 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%H A253071 Alois P. Heinz, <a href="/A253071/b253071.txt">Table of n, a(n) for n = 0..8191</a>
%H A253071 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 A253071 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 A253071 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 A253071 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 A253071 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A253071 This is the Run Length Transform of A253072.
%e A253071 Here is the neighborhood f:
%e A253071 [0, X, X]
%e A253071 [X, 0, X]
%e A253071 [X, X, X]
%e A253071 which contains a(1) = 7 ON cells.
%p A253071 C:=f->subs({x=1, y=1}, f);
%p A253071 # Find number of ON cells in CA for generations 0 thru M defined by rule
%p A253071 # that cell is ON iff number of ON cells in nbd at time n-1 was odd
%p A253071 # where nbd is defined by a polynomial or Laurent series f(x, y).
%p A253071 OddCA:=proc(f, M) global C; local n, a, i, f2, p;
%p A253071 f2:=simplify(expand(f)) mod 2;
%p A253071 a:=[]; p:=1;
%p A253071 for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
%p A253071 lprint([seq(a[i], i=1..nops(a))]);
%p A253071 end;
%p A253071 f:=1/(x*y)+1/x+1/y+y+x/y+x+x*y;
%p A253071 OddCA(f, 130);
%t A253071 (* f = A253072 *) f[0]=1; f[1]=7; f[2]=21; f[3]=95; f[4]=333; f[5]=1319; f[n_] := f[n] = -8 f[n-5] + 44 f[n-4] - 24 f[n-3] - 5 f[n-2] + 6 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* _Jean-François Alcover_, Jul 12 2017 *)
%Y A253071 Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253065, A253066, A252069.
%Y A253071 Cf. A253072.
%K A253071 nonn
%O A253071 0,2
%A A253071 _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 23 2015