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A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).

%I A246035 #39 Apr 30 2017 10:41:36
%S A246035 1,9,9,25,9,81,25,121,9,81,81,225,25,225,121,441,9,81,81,225,81,729,
%T A246035 225,1089,25,225,225,625,121,1089,441,1849,9,81,81,225,81,729,225,
%U A246035 1089,81,729,729,2025,225,2025,1089,3969,25,225,225,625,225,2025,625,3025,121,1089,1089,3025,441,3969,1849,7225,9,81,81,225,81,729,225
%N A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).
%C A246035 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 A246035 This is the odd-rule cellular automaton defined by OddRule 777 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%C A246035 Run Length Transform of {A001045(k+2)^2} (or of A139818).
%C A246035 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 A246035 Alois P. Heinz, <a href="/A246035/b246035.txt">Table of n, a(n) for n = 0..8192</a>
%H A246035 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 A246035 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 A246035 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 A246035 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 A246035 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A246035 a(n) = A071053(n)^2.
%e A246035 Here is the neighborhood:
%e A246035 [X, X, X]
%e A246035 [X, X, X]
%e A246035 [X, X, X]
%e A246035 which contains a(1) = 9 ON cells.
%e A246035 .
%e A246035 From Omar E. Pol, Mar 17 2015: (Start)
%e A246035 Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A139818(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below:
%e A246035 ..
%e A246035 9;
%e A246035 ...
%e A246035 9;
%e A246035 25;
%e A246035 ..........
%e A246035 9,     81;
%e A246035 25;
%e A246035 121;
%e A246035 ....................
%e A246035 9,     81,  81, 225;
%e A246035 25,   225;
%e A246035 121;
%e A246035 441;
%e A246035 ........................................
%e A246035 9,     81,  81, 225, 81, 729, 225, 1089;
%e A246035 25,   225, 225, 625;
%e A246035 121, 1089;
%e A246035 441;
%e A246035 1849;
%e A246035 ...
%e A246035 Note that every row r is equal to A139818(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number: T(s,r,k) = T(s+1,r,k).
%e A246035 (End)
%p A246035 C:=f->subs({x=1, y=1}, f);
%p A246035 # Find number of ON cells in CA for generations 0 thru M defined by rule
%p A246035 # that cell is ON iff number of ON cells in nbd at time n-1 was odd
%p A246035 # where nbd is defined by a polynomial or Laurent series f(x, y).
%p A246035 OddCA:=proc(f, M) global C; local n, a, i, f2, p;
%p A246035 f2:=simplify(expand(f)) mod 2;
%p A246035 a:=[]; p:=1;
%p A246035 for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
%p A246035 lprint([seq(a[i], i=1..nops(a))]);
%p A246035 end;
%p A246035 f:=(1/x+1+x)*(1/y+1+y);
%p A246035 OddCA(f, 70);
%t A246035 b[0] = 1; b[n_] := b[n] = Expand[b[n - 1]*(x^2 + x + 1)];
%t A246035 a[n_] := Count[CoefficientList[b[n], x], _?OddQ]^2;
%t A246035 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 30 2017 *)
%Y A246035 Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034.
%Y A246035 Cf. A001045, A139818.
%K A246035 nonn
%O A246035 0,2
%A A246035 _N. J. A. Sloane_, Aug 20 2014