OFFSET
0,2
LINKS
Eric Weisstein's World of Mathematics, Pi.
Eric Weisstein's World of Mathematics, Pi Continued Fraction.
FORMULA
E.g.f.: exp(asinh(x))((1+x)/(1+x^2)+(2-x+x^2)/(1+x^2)^(3/2))-2. - Michael Somos, Mar 11 2004
E.g.f.: (1+3*x+2*x^3)/(1+x^2)^(3/2). - Vaclav Kotesovec, Oct 05 2013
a(n) ~ 2*(cos(Pi*n/2)+sin(Pi*n/2)) * n^(n+1) / exp(n). - Vaclav Kotesovec, Oct 05 2013
MATHEMATICA
b[ n_ ] := 2-(-1)^n; a[ 1 ] := -1; a[ n_Integer?EvenQ ] := -n(n+1); a[ n_Integer?OddQ ] := -(n-2)(n-1); then use the standard algorithm to get p[ n ]/q[ n ].
a[n_] := Product[If[OddQ[k], k+2, 1-k], {k, 1, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 06 2012, after 1st Pari program *)
PROG
(PARI) a(n)=if(n<0, 0, prod(k=1, n, if(k%2, k+2, 1-k)))
(PARI) {a(n)=local(A); if(n<0, 0, A=matrix(2, n+1); for(k=0, n, A[2, k+1]=if(k%2, 3, 1); A[1, k+1]=if(k<2, (-1)^k, if(k%2, -(k-2)*(k-1), -k*(k+1)))); contfracpnqn(A)[2, 1])} /* Michael Somos, Jul 15 2003 */
CROSSREFS
KEYWORD
sign,frac
AUTHOR
STATUS
approved