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%I #55 Mar 09 2023 09:00:06
%S 1,6,20,88,336,1376,5440,21888,87296,349696,1397760,5593088,22368256,
%T 89481216,357908480,1431666688,5726601216,22906535936,91625881600,
%U 366504050688,1466015154176,5864062713856,23456246661120,93824995033088,375299963355136,1501199886974976,6004799480791040,24019198057381888
%N Expansion of (1+4*x)/((1+2*x)*(1-4*x)).
%C Also, fourth moments of Rudin-Shapiro polynomials (see Doche, Doche-Habsieger, Ekhad papers). - _Doron Zeilberger_, Apr 15 2016
%H Vincenzo Librandi, <a href="/A246036/b246036.txt">Table of n, a(n) for n = 0..1000</a>
%H Christophe Doche, <a href="https://doi.org/10.1090/S0025-5718-05-01736-9">Even moments of generalized Rudin-Shapiro polynomials</a>, Mathematics of computation 74.252 (2005): 1923-1935.
%H Christophe Doche and Laurent Habsieger, <a href="http://web.science.mq.edu.au/~doche/049.pdf">Moments of the Rudin-Shapiro polynomials</a>, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.
%H Shalosh B. Ekhad, <a href="https://sites.math.rutgers.edu/~zeilberg/tokhniot/oHaroldSilentShapiro1.txt">Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials</a>, Preprint, 2016.
%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_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,8).
%F a(n) = 2*a(n-1) + 8*a(n-2).
%F a(n) = (4^(1+n) - (-2)^n)/3. - _Colin Barker_, Aug 22 2014
%F a(n) = A054881(n+3)/8. - _L. Edson Jeffery_, Apr 22 2015
%F a(n) = A003683(n+2)/2 and the above formula follow from the explicit expression for a(n), cf. second formula. - _M. F. Hasler_, Sep 11 2020
%F a(n) = 2^n*A001045(n+2). - _R. J. Mathar_, Mar 08 2021
%t CoefficientList[Series[(1+4x)/((1+2x)(1-4x)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 22 2014 *)
%o (Magma) I:=[1,6]; [n le 2 select I[n] else 2*Self(n-1)+8*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 22 2014
%o (PARI) Vec((1+4*x)/((1+2*x)*(1-4*x)) + O(x^100)) \\ _Colin Barker_, Aug 22 2014
%o (PARI) apply( A246036(n)=(4^(1+n)-(-2)^n)/3, [0..30]) \\ _M. F. Hasler_, Sep 18 2020
%o (SageMath)
%o A246036= BinaryRecurrenceSequence(2,8,1,6)
%o [A246036(n) for n in range(41)] # _G. C. Greubel_, Mar 08 2023
%Y Cf. A001045, A003683, A054881, A246037, A271494, A271495, A271496.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Aug 21 2014
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