OFFSET
0,2
COMMENTS
Hankel transform is the (4,-3) Somos-4 sequence A184121.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (sqrt(2*x^2+4*x+1)-sqrt(2*x^2+1))/(2*x*sqrt(2*x^2+4*x+1)).
G.f.: 1/(1+4x+2x^2-x/(1-x/(1+4x+2x^2-x/(1-x/(1+4x+2x^2-x/(1-x/(1-... (continued fraction).
Conjecture: (n+1)*a(n) +2*(2n+1)*a(n-1) +4*(n-1)*a(n-2) +4*(2n-5)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
G.f.: 1/(2*x) - G(0)/(2*x), where G(k)= 1 - 4*x*(4*k+1)/( (1+2*x^2)*(4*k+2) - x*(1+2*x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) - (1+2*x^2)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) ~ (-1)^n * (2+sqrt(2))^n / (sqrt(3*sqrt(2)-4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014
MATHEMATICA
CoefficientList[Series[(Sqrt[2x^2+4x+1]-Sqrt[2x^2+1])/(2x Sqrt[2x^2+4x+1]), {x, 0, 30}], x] (* Harvey P. Dale, Mar 09 2012 *)
PROG
(PARI) x='x+O('x^30); Vec((sqrt(2*x^2+4*x+1)-sqrt(2*x^2+1))/( 2*x*sqrt(2*x^2+4*x+1))) \\ G. C. Greubel, Aug 14 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((Sqrt(2*x^2+4*x+1)-Sqrt(2*x^2+1))/( 2*x*Sqrt(2*x^2+4*x+1)))); // G. C. Greubel, Aug 14 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Barry, Jan 09 2011
STATUS
approved