| G.f.: x*(1-x)/(1+x)/(1-6x+x^2). a(n)=(r^n+(1/r)^n-2*(-1)^n)/8, with r=3+sqrt(8). a(n+3)=5*a(n+2)+5*a(n+1)-a(n).
L.g.f.: 1/8*log((1+2*x+x^2)/(1-6*x+x^2)) = sum(n>=0, a(n)/n*x^n), see p.627 of the fxtbook link; special case of the following: let v(0)=0, v(1)=1, and v(n)=u*v(n-1)+v(n-2), then 1/A*log((1+2*x+x^2)/(1-(2-A)*x+x^2)) = sum(n>=0, v(n)^2/n*x^n) where A=u^2+4. [Joerg Arndt, Apr 08 2011]
a(n+1) = sum_{k=0...n}((-1)^(n-k)*A001653(k)); e.g. 144 = -1 + 5 - 29 + 169; 25 = 1 - 5 + 29 - Charlie Marion (charliem(AT)bestweb.net), Jul 16 2003
a(n)=A000129(n)^2.
a(n)= (T(n, 3)-(-1)^n)/4 with Chebyshev's polynomials of the first kind evaluated at x=3: T(n, 3)=A001541(n)=((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)/2. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
a(n) = rightmost term of M^n * [1 0 0] where M = the 3 X 3 matrix [4 4 1 / 2 1 0 / 1 0 0]. a(n+1) = leftmost term. E.g. a(6) = 4900, a(5) = 841 since M^5 * [1 0 0] = [4900 2030 841]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 31 2004
a(n) = [(-1)^(n+1)+A001109(n+1)-3*A001109(n)]/4. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
a(n) = ( ( ( (1-Sqrt[ 2 ])^n + (1+Sqrt[ 2 ])^n) /2 )^2 + (-1)^(n+1) ) /2 - Antonio Pane (apane1(AT)spc.edu), Dec 15 2007
Limit(a(n+k)/a(k),k=infinity) = A001541(n)+2*A001109(n)*sqrt(2). - Johannes W. Meijer, Aug 01 2010
For n>0, a(2n) = a(2n-1)-a(2n-2)-2, a(2n+1) = a(2n)-a(2n-1)+2. - Charlie Marion, Sep 24 2011
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