%I #46 Feb 05 2024 01:01:13
%S 0,1,1,9,25,121,441,1849,7225,29241,116281,466489,1863225,7458361,
%T 29822521,119311929,477204025,1908903481,7635439161,30542106169,
%U 122167725625,488672300601,1954686406201,7818751217209,31274993684025
%N Squares of Jacobsthal numbers.
%C Run length transform gives A246035. - _N. J. A. Sloane_, Feb 26 2015
%H Vincenzo Librandi, <a href="/A139818/b139818.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, 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 (3,6,-8).
%F a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3).
%F a(n) = (A001045(n))^2.
%F G.f.: x*(1-2*x)/((1-x)*(1+2*x)*(1-4*x)).
%t LinearRecurrence[{3, 6, -8}, {0, 1, 1}, 25] (* _Jean-François Alcover_, Jan 09 2019 *)
%o (Magma) [1/9-(2/9)*(-2)^n+(1/9)*4^n: n in [0..35]]; // _Vincenzo Librandi_, Aug 09 2011
%o (PARI) concat (0, Vec(x*(1-2*x)/((1-x)*(1+2*x)*(1-4*x)) + O(x^30))) \\ _Michel Marcus_, Mar 04 2015
%Y Cf. A001045, A246035. First differences give (apart from signs) A083086.
%K nonn,easy
%O 0,4
%A _Paul Curtz_, May 17 2008
%E More terms from _R. J. Mathar_, Dec 12 2009