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a(n) = A255473(2^n-1).
2

%I #11 Feb 05 2017 06:43:57

%S 1,6,24,88,336,1280,4928,19072,74240,290304,1139712,4489216,17731584,

%T 70197248,278429696,1106083840,4399628288,17518559232,69815500800,

%U 278424715264,1110989340672,4435189170176,17712382214144,70757707153408,282733687341056,1129973180006400

%N a(n) = A255473(2^n-1).

%H Colin Barker, <a href="/A255474/b255474.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_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4,-16).

%F G.f.: (1-8*x^2-16*x^3) / ((1-4*x)*(1-2*x-4*x^2)).

%F From _Colin Barker_, Feb 05 2017: (Start)

%F a(n) = 4^n + (-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n)) / (2*sqrt(5)) for n>0.

%F a(n) = 6*a(n-1) - 4*a(n-2) - 16*a(n-3) for n>3.

%F (End)

%o (PARI) Vec((1-8*x^2-16*x^3) / ((1-4*x)*(1-2*x-4*x^2)) + O(x^30)) \\ _Colin Barker_, Feb 05 2017

%Y Cf. A255473.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 23 2015