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a(n) = A255458(2^n-1).
3

%I #15 Jan 09 2019 04:04:09

%S 1,5,25,101,361,1205,3865,12101,37321,114005,346105,1046501,3155881,

%T 9500405,28566745,85831301,257756041,773792405,2322425785,6969374501,

%U 20912317801,62745342005,188252803225,564791964101,1694443001161,5083463221205,15250658099065

%N a(n) = A255458(2^n-1).

%H Colin Barker, <a href="/A255459/b255459.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,-11,6).

%F G.f.: (1-x+6*x^2) / ((1-x)*(1-2*x)*(1-3*x)).

%F From _Colin Barker_, Feb 03 2017: (Start)

%F a(n) = (3 - 2^(3+n) + 2*3^(1+n)).

%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>2.

%F (End)

%t LinearRecurrence[{6, -11, 6}, {1, 5, 25}, 30] (* _Jean-François Alcover_, Jan 09 2019 *)

%o (PARI) Vec((1-x+6*x^2) / ((1-x)*(1-2*x)*(1-3*x)) + O(x^30)) \\ _Colin Barker_, Feb 03 2017

%Y Cf. A255458.

%K nonn,easy

%O 0,2

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