|
%I #19 Feb 04 2017 10:42:47
%S 1,5,15,51,153,477,1431,4347,13041,39285,117855,354051,1062153,
%T 3187917,9563751,28695627,86086881,258273765,774821295,2324503251,
%U 6973509753,20920647357,62761942071,188286180507,564858541521,1694576687445,5083730062335,15251193375651,45753580126953,137260749946797
%N a(n) = A255304(2^n-1).
%H Colin Barker, <a href="/A255442/b255442.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 (3,3,-9).
%F G.f.: (1+3*x)*(1-x) / ((1-3*x)*(1-3*x^2)).
%F From _Colin Barker_, Feb 04 2017: (Start)
%F a(n) = 2*3^n - 3^((n-1)/2)/2*(1-(-1)^n+sqrt(3) + (-1)^n*sqrt(3)).
%F a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3) for n>2.
%F (End)
%t A255442[n_] := SeriesCoefficient[(1 + 3*x)*(1 - x)/((1 - 3*x)*(1 - 3*x^2)), {x, 0, n}]; Array[A255442, 30, 0] (* _JungHwan Min_, Sep 29 2016 *)
%t A255442L[n_] := CoefficientList[Series[(1 + 3*x)*(1 - x)/((1 - 3*x)*(1 - 3*x^2)), {x, 0, n}], x]; A255442L[29] (* _JungHwan Min_, Sep 29 2016 *)
%o (PARI) Vec((1+3*x)*(1-x) / ((1-3*x)*(1-3*x^2)) + O(x^30)) \\ _Colin Barker_, Feb 04 2017
%Y Cf. A255304.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 23 2015
|
