OFFSET
0,3
COMMENTS
Denominators 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-...))))))). - Michael Somos, Aug 07 2000
Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, ...) as the main diagonal and (-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
FORMULA
a(n) = -(-1)^n*A053984(-1-n). - Michael Somos, Aug 07 2000
E.g.f.: cos(1-sqrt(1-2*x))/sqrt(1-2*x). If a(0)=0, a(n)=0, 1, 1, 2, 9, 61, 540, 5879, 75887, 1132426, ... then e.g.f. = sin(1)*cos(sqrt(1-2*x))-cos(1)*sin(sqrt(1-2*x)). - Miklos Kristof, Jun 15 2005, corrected by Vaclav Kotesovec, Jul 31 2014
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*(n-2*k)!*binomial(n-k,k) * binomial(n-k-1/2,k-1/2). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) ~ cos(1) * 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Jul 31 2014
a(n) = 2^n*Gamma(n+1/2)*hypergeometric([1/2-n/2, -n/2], [1/2, 1/2-n, -n], -1)/sqrt(Pi) 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
(1/(2^n*n!)) * Integral_{x = 0..1} (1 - x^2)^n*cos(x) dx = a(n)*sin(1) - A053984(n)*cos(1). Hence A053984(n)/a(n) -> tan(1) as n -> infinity. - Peter Bala, Mar 06 2015
a(n) = SphericalBesselJ[0,1]*SphericalBesselJ[n,1] + SphericalBesselY[0,1]*SphericalBesselY[n,1]. - G. C. Greubel, May 10 2015
Sum_{n>0} a(n-1) t^n/n! = sin(1 - sqrt(1-2t)). - G. C. Greubel, May 10 2015
EXAMPLE
a(10) = 363199319 because 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/(15-1/(17-1/19))))))))) = 565649425/363199319.
MAPLE
E(x):=sin(1)*cos(sqrt(1-2*x))-cos(1)*sin(sqrt(1-2*x)): f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1], x) od: x:=0: for n from 1 to 30 do f[n]:=simplify(f[n]/(sin(1)^2+cos(1)^2)) od: seq(f[n], n=1..30); # Miklos Kristof, Jun 15 2005
MATHEMATICA
RecurrenceTable[{a[0]==a[1]==1, a[n]==(2n-1)a[n-1]-a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Dec 21 2011 *)
CoefficientList[Series[Cos[1-Sqrt[1-2*x]]/Sqrt[1-2*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 31 2014 *)
PROG
(Sage)
def A053983(n):
if n < 2: return 1
return 2^n*gamma(n+1/2)*hypergeometric([1/2-n/2, -n/2], [1/2, 1/2-n, -n], -1)/sqrt(pi)
[round(A053983(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 10 2014
(PARI) a(n)={if(n<2, 1, (2*n-1)*a(n-1)-a(n-2))} \\ Edward Jiang, Sep 10 2014
(PARI) {a(n) = my(a0, a1, s=n<0); if( n>-3 && n<1, return(n+1)); if( n<0, n=-1-n); a0=1-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) [1] cat [ n le 2 select n else (2*n-1)*Self(n-1)-Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Mar 08 2015
CROSSREFS
KEYWORD
easy,frac,nonn
AUTHOR
Vladeta Jovovic, Apr 02 2000
EXTENSIONS
Additional comments from Michael Somos, Aug 23 2000
More terms from Miklos Kristof, Jun 15 2005
STATUS
approved