OFFSET
0,3
COMMENTS
Numerators of successive convergents to tan(1) using continued fraction 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/15-...))))))).
Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, 7, ...) as the main diagonal and (0, -1, -1, -1, ...) as the subdiagonal. - Gary W. Adamson, Apr 20 2009
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
FORMULA
a(n) = (-1)^n*A053983(-1-n). - Michael Somos, Aug 23 2000 [See Somos's formula in A053983 which is valid for all n in Z.]
E.g.f.: sin(1-sqrt(1-2*x))/sqrt(1-2*x). Cf. A036244. - Vladeta Jovovic, Aug 10 2006
Recurrence equation: a(n+1) = (2*n+1)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k-1/2,k+1/2), cf. A058798. - Peter Bala, Aug 01 2013
a(n) ~ sin(1)*2^(n+1/2)*n^n/exp(n). - Vaclav Kotesovec, Oct 05 2013
a(n) = (2*n-1)!!*hypergeometric([1 - n/2, 1/2 - n/2], [3/2, 1 - n, 1/2 - n], -1) for n >= 2. - Peter Luschny, Sep 10 2014
0 = a(n)*(+a(n+2)) + a(n+1)*(-a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z. - Michael Somos, Sep 11 2014
a(n) = SphericalBesselJ[n,1]*SphericalBesselY[0,1] - SphericalBesselJ[0,1]*SphericalBesselY[n,1]. - G. C. Greubel, May 10 2015
Sum_{n>=0} a(n-1)*t^n/n! = - cos(1 - sqrt(1-2*t)), where a(-1) = -1. - G. C. Greubel, May 10 2015
The SphericalBessel formula given by Greubel above can be rewritten as a(n) = sqrt(Pi/2)*(-cos(1)*BesselJ(n+1/2, 1) + (-1)^n*sin(1)*BesselJ(-(n+1/2), 1)). - Wolfdieter Lang, Jun 14 2015
EXAMPLE
a(10)=565649425 because 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/(15-1/(17-1/19))))))))) = 565649425/363199319.
MAPLE
f:= gfun:-rectoproc({a(n)=(2*n-1)*a(n-1)-a(n-2), a(0)=0, a(1)=1}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 14 2015
MATHEMATICA
CoefficientList[Series[Sin[1-Sqrt[1-2*x]]/Sqrt[1-2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)
RecurrenceTable[{a[n] == (2*n - 1)*a[n - 1] - a[n - 2], a[0] == 0,
a[1] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jan 22 2017 *)
PROG
(Sage)
def A053984(n):
if n < 2: return n
return 2^n*gamma(n+1/2)*hypergeometric([1-n/2, 1/2-n/2], [3/2, 1 - n, 1/2 -n], -1) / sqrt(pi)
[round(A053984(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 10 2014
(PARI) a(n)={if(n<2, n, (2*n-1)*a(n-1)-a(n-2))} \\ Edward Jiang, Sep 10 2014
(PARI) {a(n) = my(a0, a1, s=n<0); if( abs(n) < 2, return(n)); if( n<0, n=-1-n); a0=s; a1=1; for(k=2, n, a2 = (2*k-1)*a1 - a0; a0=a1; a1=a2); (-1)^(s*n) * a1}; /* Michael Somos, Sep 11 2014 */
(Magma) [n le 2 select (n-1) else (2*n-3)*Self(n-1)-Self(n-2): n in [1..25] ]; // Vincenzo Librandi, May 12 2015
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Vladeta Jovovic, Apr 02 2000
EXTENSIONS
Additional comments from Michael Somos, Aug 23 2000
More terms from Vladeta Jovovic, Aug 10 2006
STATUS
approved