A191993 a(n) = 3^(n-1) + C(2*n, n)/2.
2, 6, 19, 62, 207, 705, 2445, 8622, 30871, 112061, 411765, 1529225, 5731741, 21652623, 82341729, 314889102, 1209849831, 4666707813, 18060052389, 70085525877, 272615721621, 1062509835063, 4148096423409, 16217945020377, 63487732755357, 248806555083495
Offset: 1
Examples
a(5) = 3^4 + C(10,5)/2 = 81 + 126 = 207.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Programs
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Maple
seq(3^(n-1)+binomial(2*n-1,n),n=1..20)
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Mathematica
Table[3^(n-1)+Binomial[2n,n]/2,{n,30}] (* Harvey P. Dale, Dec 27 2011 *) -
PARI
a(n)=3^(n-1)+binomial(n+n,n)/2 \\ Charles R Greathouse IV, Jun 21 2011
Formula
a(n) = Sum_{k=0..floor(n/3)} (-1)^k*C(2*n, n-3*k).
G.f.: ((x-1)*(4*x-1) + sqrt((1-4*x)*(3*x-1)^2))/(2*(4*x-1)*(3*x-1)) - 1.
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.g.f.: (exp(2*x)*(2*exp(x) + 3*BesselI(0,2*x)) - 5)/6. - Stefano Spezia, Aug 22 2025