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a(n) = A255466(2^n-1).
2

%I #14 Feb 05 2017 03:54:18

%S 1,6,26,110,450,1822,7330,29406,117794,471518,1886754,7548382,

%T 30196258,120790494,483172898,1932713438,7730897442,30923677150,

%U 123694883362,494779882974,1979120230946,7916482321886,31665932083746,126663733927390,506654946894370,2026619809947102,8106479284527650,32425917227589086

%N a(n) = A255466(2^n-1).

%H Colin Barker, <a href="/A255467/b255467.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 (5,-2,-8).

%F G.f.: (1+2*x)*(1-x) / ((1-4*x)*(1-2*x)*(1+x)).

%F From _Colin Barker_, Feb 04 2017: (Start)

%F a(n) = (-2*(-1)^n/15 - 2^(1+n)/3 + (9*4^n)/5).

%F a(n) = 5*a(n-1) - 2*a(n-2) - 8*a(n-3) for n>2.

%F (End)

%t CoefficientList[Series[(1 + 2*x)*(1 - x)/((1 - 4*x)*(1 - 2*x)*(1 + x)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Feb 04 2017 *)

%o (PARI) Vec((1+2*x)*(1-x) / ((1-4*x)*(1-2*x)*(1+x)) + O(x^30)) \\ _Colin Barker_, Feb 04 2017

%Y Cf. A255466.

%K nonn,easy

%O 0,2

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