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%I #16 Oct 10 2018 03:24:26
%S 1,4,16,56,196,680,2348,8096,27892,96056,330748,1138768,3920644,
%T 13498088,46471180,159990272,550811156,1896319640,6528602140,
%U 22476505520,77381536036,266407155784,917179667500,3157642420064,10871049557044,37426567849976,128851218332732,443605636686608,1527233994485572
%N a(n) = A255300(2^k-1).
%H Colin Barker, <a href="/A255301/b255301.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 [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>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-2,-4).
%F G.f.: (1-x)*(1+x+2*x^2) / (1-4*x+x^2+2*x^3+4*x^4).
%F a(n) = 4*a(n-1) - a(n-2) - 2*a(n-3) - 4*a(n-4) for n>3. - _Colin Barker_, Feb 04 2017
%t LinearRecurrence[{4, -1, -2, -4}, {1, 4, 16, 56}, 30] (* _Jean-François Alcover_, Oct 10 2018 *)
%o (PARI) Vec((1-x)*(1+x+2*x^2) / (1-4*x+x^2+2*x^3+4*x^4) + O(x^30)) \\ _Colin Barker_, Feb 04 2017
%Y Cf. A255300.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 23 2015
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