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A246030 a(n) = (5*2^(2*n)+(-2)^(n+1))/3. 3

%I

%S 1,8,24,112,416,1728,6784,27392,109056,437248,1746944,6991872,

%T 27959296,111853568,447381504,1789591552,7158235136,28633202688,

%U 114532286464,458130194432,1832518680576,7330078916608,29320307277824,117281245888512,469124949999616,1876499867107328,7505999334211584

%N a(n) = (5*2^(2*n)+(-2)^(n+1))/3.

%C Essentially 8 times A083424.

%C Equals A160239(2^n). - _N. J. A. Sloane_, Oct 04 2014

%H Vincenzo Librandi, <a href="/A246030/b246030.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, 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, 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, 2015

%H N. J. A. Sloane, <a href="/A160239/a160239_2.pdf">Illustration for a(4) = 416.</a>

%H N. J. A. Sloane, <a href="/A160239/a160239_1.png">Illustration for a(5) = 1728.</a>

%H N. J. A. Sloane, <a href="/A160239/a160239_4.png">Illustration for a(6) = 6784.</a>

%H N. J. A. Sloane, <a href="/A160239/a160239_1.tiff">Illustration for a(7) = 27392 (tiff).</a>

%H N. J. A. Sloane, <a href="/A160239/a160239_5.png">Illustration for a(7) = 27392 (png).</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,8).

%F a(0)=1, a(1)=8; thereafter a(n)=2*a(n-1)+8*a(n-2).

%F G.f.: (1+6*x)/(1-2*x-8*x^2).

%t CoefficientList[Series[(1 + 6 x) / (1 - 2 x - 8 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 20 2015 *)

%o (MAGMA) I:=[1,8]; [n le 2 select I[n] else 2*Self(n-1)+8*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 20 2015

%Y Cf. A083424, A160239.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Aug 15 2014

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Last modified November 19 22:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)