This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A166956 #26 Feb 16 2025 08:33:11
%S A166956 0,-1,3,5,15,29,63,125,255,509,1023,2045,4095,8189,16383,32765,65535,
%T A166956 131069,262143,524285,1048575,2097149,4194303,8388605,16777215,
%U A166956 33554429,67108863,134217725,268435455,536870909,1073741823,2147483645,4294967295,8589934589
%N A166956 a(n) = 2^n +(-1)^n - 2.
%C A166956 The inverse binomial transform yields 0,-1,5,-7,17,-31,..., a sign alternating variant of A014551.
%C A166956 In a table of a(n) and higher-order differences in successive rows, the main diagonal contains 0, 4, 8, 16, ... (zero followed by A020707).
%C A166956 Similar to 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 899", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero, which begins with 1,3,5,15,29,63,125. - _Robert Price_, Aug 08 2017
%D A166956 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A166956 Vincenzo Librandi, <a href="/A166956/b166956.txt">Table of n, a(n) for n = 0..240</a>
%H A166956 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 A166956 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A166956 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A166956 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A166956 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A166956 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A166956 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A166956 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2)
%F A166956 a(n) = A000079(n) - A010684(n).
%F A166956 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
%F A166956 G.f.: x*(5*x -1)/((1-x)*(1-2*x)*(1+x)).
%F A166956 E.g.f.: exp(2*x) - 2*exp(x) + exp(-x). - _G. C. Greubel_, May 29 2016
%t A166956 LinearRecurrence[{2,1,-2},{0,-1,3},20] (* _G. C. Greubel_, May 29 2016 *)
%o A166956 (Magma) [2^n-2+(-1)^n: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011
%Y A166956 Cf. A166920, A166956, A290660, A290661, A290662.
%K A166956 sign,easy
%O A166956 0,3
%A A166956 _Paul Curtz_, Oct 25 2009
%E A166956 Edited and extended by _R. J. Mathar_, Mar 02 2010