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 A164908 #44 Mar 31 2025 16:57:33
%S A164908 1,6,24,96,384,1536,6144,24576,98304,393216,1572864,6291456,25165824,
%T A164908 100663296,402653184,1610612736,6442450944,25769803776,103079215104,
%U A164908 412316860416,1649267441664,6597069766656,26388279066624,105553116266496,422212465065984,1688849860263936
%N A164908 a(n) = (3*4^n - 0^n)/2.
%C A164908 Binomial transform of A164907. Inverse binomial transform of A057651.
%C A164908 Partial sums are in A083420.
%C A164908 Decimal representations of the n-th iterations of elementary cellular automata rules 14, 46, 142 and 174 generate this sequence (see A266298 and A266299). - _Karl V. Keller, Jr._, Aug 31 2021
%H A164908 Vincenzo Librandi, <a href="/A164908/b164908.txt">Table of n, a(n) for n = 0..1000</a>
%H A164908 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 A164908 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 A164908 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 A164908 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 A164908 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.
%H A164908 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F A164908 a(n) = 4*a(n-1) for n > 1; a(0) = 1, a(1) = 6.
%F A164908 G.f.: (1+2*x)/(1-4*x).
%F A164908 a(n) = floor(6*4^(n-1)). - _Karl V. Keller, Jr._, Aug 30 2021
%F A164908 E.g.f.: (3*exp(4*x) - 1)/2. - _Elmo R. Oliveira_, Mar 31 2025
%t A164908 a[n_]:=(MatrixPower[{{2,2},{2,2}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t A164908 Join[{1},(3*4^Range[25])/2] (* or *) Join[{1},NestList[4#&,6,25]] (* _Harvey P. Dale_, Feb 14 2012 *)
%o A164908 (Magma) [ (3*4^n-0^n)/2: n in [0..22] ];
%o A164908 (PARI) a(n)=3*4^n\2 \\ _Charles R Greathouse IV_, Oct 12 2015
%o A164908 (Python) print([int(6*4**(n-1)) for n in range(50)]) # _Karl V. Keller, Jr._, Aug 30 2021
%Y A164908 Equals 1 followed by A002023 (6*4^n). Essentially the same as A084509.
%Y A164908 Cf. A164907, A057651, A083420 (2*4^n-1), A247640, A266298, A266299.
%K A164908 nonn,easy
%O A164908 0,2
%A A164908 _Klaus Brockhaus_, Aug 31 2009