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A253101 a(n) = A253100(2^n-1). 2

%I #34 Jan 09 2019 03:54:32

%S 1,6,24,96,372,1416,5340,20040,75012,280392,1047324,3910440,14597508,

%T 54485736,203357724,758969736,2832570372,10571410056,39453266460,

%U 147242049000,549515715972,2050822387752,7653776980764,28564291826760,106603402909188,397849344975816,1484794027325724,5541326864990376

%N a(n) = A253100(2^n-1).

%H Colin Barker, <a href="/A253101/b253101.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,-9,2).

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

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

%F a(n) = -2^n + 2*(2-sqrt(3))^n + 2*(2+sqrt(3))^n for n>0.

%F a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) for n>3.

%F (End)

%t Join[{1}, LinearRecurrence[{6, -9, 2}, {6, 24, 96}, 30]] (* _Jean-François Alcover_, Jan 09 2019 *)

%o (PARI) Vec((1-3*x^2+4*x^3) / ((1-2*x)*(1-4*x+x^2)) + O(x^30)) \\ _Colin Barker_, Feb 04 2017

%Y Cf. A253100.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015

%E Typo in the definition corrected by _Antti Karttunen_, Nov 18 2016

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)