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Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).
13

%I #39 Apr 30 2017 10:41:36

%S 1,9,9,25,9,81,25,121,9,81,81,225,25,225,121,441,9,81,81,225,81,729,

%T 225,1089,25,225,225,625,121,1089,441,1849,9,81,81,225,81,729,225,

%U 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 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+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 777 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

%C Run Length Transform of {A001045(k+2)^2} (or of A139818).

%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="/A246035/b246035.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>

%F a(n) = A071053(n)^2.

%e Here is the neighborhood:

%e [X, X, X]

%e [X, X, X]

%e [X, X, X]

%e which contains a(1) = 9 ON cells.

%e .

%e From Omar E. Pol, Mar 17 2015: (Start)

%e 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 ..

%e 9;

%e ...

%e 9;

%e 25;

%e ..........

%e 9, 81;

%e 25;

%e 121;

%e ....................

%e 9, 81, 81, 225;

%e 25, 225;

%e 121;

%e 441;

%e ........................................

%e 9, 81, 81, 225, 81, 729, 225, 1089;

%e 25, 225, 225, 625;

%e 121, 1089;

%e 441;

%e 1849;

%e ...

%e 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 (End)

%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+1+y);

%p OddCA(f, 70);

%t b[0] = 1; b[n_] := b[n] = Expand[b[n - 1]*(x^2 + x + 1)];

%t a[n_] := Count[CoefficientList[b[n], x], _?OddQ]^2;

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 30 2017 *)

%Y Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034.

%Y Cf. A001045, A139818.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Aug 20 2014