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%I #24 Mar 15 2023 14:43:03
%S 2,6,19,62,207,705,2445,8622,30871,112061,411765,1529225,5731741,
%T 21652623,82341729,314889102,1209849831,4666707813,18060052389,
%U 70085525877,272615721621,1062509835063,4148096423409,16217945020377,63487732755357,248806555083495
%N a(n) = 3^(n-1) + C(2*n, n)/2.
%H G. C. Greubel, <a href="/A191993/b191993.txt">Table of n, a(n) for n = 1..1000</a>
%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
%F a(n) = A000244(n-1) + A001700(n-1).
%F a(n) = Sum_{k=0..floor(n/3)} (-1)^k*C(2*n, n-3*k).
%F G.f.: ((x-1)*(4*x-1) + sqrt((1-4*x)*(3*x-1)^2))/(2*(4*x-1)*(3*x-1)) - 1.
%F Conjecture: n*(n-3)*a(n) - (7*n^2 -23*n +12)*a(n-1) +6*(2*n-3)*(n-2)*a(n-2)=0. - _R. J. Mathar_, Oct 18 2017
%e a(5) = 3^4 + C(10,5)/2 = 81 + 126 = 207.
%p seq(3^(n-1)+binomial(2*n-1,n),n=1..20)
%t Table[3^(n-1)+Binomial[2n,n]/2,{n,30}] (* _Harvey P. Dale_, Dec 27 2011 *)
%o (PARI) a(n)=3^(n-1)+binomial(n+n,n)/2 \\ _Charles R Greathouse IV_, Jun 21 2011
%Y Cf. A000244, A001700, A005317.
%K nonn,easy
%O 1,1
%A _Mircea Merca_, Jun 21 2011
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