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a(n) = (3*4^n - 0^n)/2.
9

%I #39 Sep 08 2022 08:45:47

%S 1,6,24,96,384,1536,6144,24576,98304,393216,1572864,6291456,25165824,

%T 100663296,402653184,1610612736,6442450944,25769803776,103079215104,

%U 412316860416,1649267441664,6597069766656,26388279066624,105553116266496,422212465065984

%N a(n) = (3*4^n - 0^n)/2.

%C Binomial transform of A164907. Inverse binomial transform of A057651.

%C Partial sums are in A083420.

%C 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 Vincenzo Librandi, <a href="/A164908/b164908.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 [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_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).

%F a(n) = 4*a(n-1) for n > 1; a(0) = 1, a(1) = 6.

%F G.f.: (1+2*x)/(1-4*x).

%F a(n) = floor(6*4^(n-1)). - _Karl V. Keller, Jr._, Aug 30 2021

%t 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 Join[{1},(3*4^Range[25])/2] (* or *) Join[{1},NestList[4#&,6,25]] (* _Harvey P. Dale_, Feb 14 2012 *)

%o (Magma) [ (3*4^n-0^n)/2: n in [0..22] ];

%o (PARI) a(n)=3*4^n\2 \\ _Charles R Greathouse IV_, Oct 12 2015

%o (Python) print([int(6*4**(n-1)) for n in range(50)]) # _Karl V. Keller, Jr._, Aug 30 2021

%Y Equals 1 followed by A002023 (6*4^n). Essentially the same as A084509.

%Y Cf. A164907, A057651, A083420 (2*4^n-1), A247640, A266298, A266299.

%K nonn,easy

%O 0,2

%A _Klaus Brockhaus_, Aug 31 2009