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%I #63 Nov 27 2017 02:49:27
%S 1,6,22,82,302,1106,4066,14902,54678,200578,735770,2699182,9901550,
%T 36323050,133247570,488805718,1793137798,6577952882,24130592458,
%U 88520767614,324729961566,1191240790586,4369952806274,16030753627238,58807285300086,215728897446594,791380812129402,2903104763095054
%N The subsequence A253069(2^n-1).
%C A253069 is the Run Length Transform of this sequence.
%H Colin Barker, <a href="/A253070/b253070.txt">Table of n, a(n) for n = 0..1000</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, 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, 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, 2015
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-1,-6,10,-8,8).
%F G.f.: (1+2*x)*(1+x-x^2+x^3+2*x^5)/(1-3*x-3*x^2+x^3+6*x^4-10*x^5+8*x^6-8*x^7). - _Doron Zeilberger_, Feb 18 2015
%p OddCA2:=proc(f,M) local n,a,i,f2,g,p;
%p f2:=simplify(expand(f)) mod 2;
%p p:=1; g:=f2;
%p for n from 1 to M do p:=expand(p*g) mod 2; print(n,nops(p)); g:=expand(g^2) mod 2; od:
%p return;
%p end;
%p f:=1/x+1+x+x/y+y/x+x*y;
%p OddCA2(f,10);
%t LinearRecurrence[{3, 3, -1, -6, 10, -8, 8}, {1, 6, 22, 82, 302, 1106, 4066}, 28] (* _Jean-François Alcover_, Nov 27 2017 *)
%o (PARI) Vec(-(2*x+1)*(2*x^5+x^3-x^2+x+1)/(8*x^7-8*x^6+10*x^5-6*x^4-x^3+3*x^2+3*x-1) + O(x^30)) \\ _Colin Barker_, Jul 16 2015
%Y Cf. A253067, A253068, A253069.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 29 2015
%E a(11) and a(12) (Maple on a 32 GB machine) from _R. J. Mathar_, Feb 04 2015
%E a(13) onwards from _Doron Zeilberger_, Feb 18 2015 (the terms previously listed were wrong). - _N. J. A. Sloane_, Feb 19 2015
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