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Expansion of (1-x)^(-1)/(1+x-2*x^2-2*x^3).
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%I #16 Sep 08 2022 08:45:07

%S 1,0,3,0,7,0,15,0,31,0,63,0,127,0,255,0,511,0,1023,0,2047,0,4095,0,

%T 8191,0,16383,0,32767,0,65535,0,131071,0,262143,0,524287,0,1048575,0,

%U 2097151,0,4194303,0,8388607,0,16777215,0,33554431,0,67108863,0,134217727,0,268435455

%N Expansion of (1-x)^(-1)/(1+x-2*x^2-2*x^3).

%C Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 276", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - _Robert Price_, May 25 2017

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

%H Vincenzo Librandi, <a href="/A077896/b077896.txt">Table of n, a(n) for n = 0..1000</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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>

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

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-2).

%F G.f.: 1/((1 - x)*(1 + x)*(1 - 2*x^2)). - _Bruno Berselli_, May 26 2017

%F a(n) = (1 + (-1)^n)*(2^floor((n + 3)/2) - 1)/2. - _Vincenzo Librandi_, May 27 2017

%t CoefficientList[Series[(1 - x)^(-1) / (1 + x - 2 x^2 - 2 x^3), {x, 0, 60}], x] (* _Vincenzo Librandi_, May 26 2017 *)

%o (Magma) [(1+(-1)^n)*(2^Floor((n+3)/2)-1)/2: n in [0..60]]; // _Vincenzo Librandi_, May 26 2017

%Y Cf. A287468, A287469, A287470.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Nov 17 2002