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%I #79 Dec 31 2022 13:21:45
%S 1,1,3,3,9,9,27,27,81,81,243,243,729,729,2187,2187,6561,6561,19683,
%T 19683,59049,59049,177147,177147,531441,531441,1594323,1594323,
%U 4782969,4782969,14348907,14348907,43046721,43046721,129140163,129140163,387420489,387420489,1162261467
%N a(n) = 3^floor(n/2). Powers of 3 repeated.
%C a(n) is the Parker sequence for the automorphism group of the limit of the class of oriented graphs; a(n) counts the finite circulant structures in that class. - _N-E. Fahssi_, Feb 18 2008
%C Complete sequence: every positive integer is the sum of members of this sequence. - _Charles R Greathouse IV_, Jul 19 2012
%C Conjecture: a(n+1) is the number of distinct subsets S of {0,1,2,...,n} such that the sumset S+S does not contain n. - _Michael Chu_, Oct 05 2021. _Andrew Howroyd_, Nov 20 2021: The conjecture is true: If there are m pairs of numbers that add to n then inclusion/exclusion gives sum(k=0, m, binomial(m,k)*(-1)^k*2^(2*m-2*k)) as the number of sets that don't contain any of those pairs which equals 3^m. For even n , n/2 cannot be included in any set.
%H Vincenzo Librandi, <a href="/A108411/b108411.txt">Table of n, a(n) for n = 0..3000</a>
%H D. A. Gewurz and F. Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seq., 6 (2003), 03.1.6.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F O.g.f.: (1+x)/(1-3*x^2). - _R. J. Mathar_, Apr 01 2008
%F a(n) = 3^(n/2)*((1+(-1)^n)/2+(1-(-1)^n)/(2*sqrt(3))). - _Paul Barry_, Nov 12 2009
%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011
%F a(n) = (-1)^n*sum(A158020(n,k)*2^k, 0<=k<=n). - _Philippe Deléham_, Dec 01 2011
%F a(n) = sum(A152815(n,k)*2^k, 0<=k<=n). - _Philippe Deléham_, Apr 22 2013
%F a(n) = 3^A004526(n). - _Michel Marcus_, Aug 30 2014
%F E.g.f.: cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3). - _Stefano Spezia_, Dec 31 2022
%e a(6) = 27; 3^floor(6/2) = 3^floor(3) = 3^3 = 27.
%p A108411:=n->3^floor(n/2); seq(A108411(k), k=0..100); # _Wesley Ivan Hurt_, Nov 01 2013
%t Table[3^Floor[n/2], {n,0,100}] (* _Wesley Ivan Hurt_, Nov 01 2013 *)
%o (PARI) a(n)=3^floor(n/2);
%o (Magma) [3^Floor(n/2): n in [0..50]]; // _Vincenzo Librandi_, Aug 17 2011
%o (Haskell)
%o a108411 = (3 ^) . flip div 2 -- _Reinhard Zumkeller_, May 01 2014
%Y Essentially the same as A056449 and A162436.
%Y Cf. A000244, A004526, A016116, A152815, A158020.
%K nonn,easy
%O 0,3
%A _Ralf Stephan_, Jun 05 2005
%E Incorrect formula removed by _Michel Marcus_, Oct 06 2021
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