0,3

The corresponding denominators are A001879(n), n>=0.

Pi= 3 + 1^2/(6+3^2/(6+5^2/(6+... See the J.-P. Delahaye reference. R. Rosenthal mentioned this continued fraction in an e-mail to the author Jul 16 2008.

For the approximants in lowest terms cf. the ones for 3*(Pi-3) given by A130411(n)/A130412(n) in lowest terms.

J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. In German: Pi - die Story, Birkhaeuser, 1999 Basel, p. 87.

W. Lang, Approximants for Pi-3 and more.

a(n)= 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(0)=0, a(1)=1.

E.g.f.: (-3*(1+x-sqrt(1-4*x^2))+ 2*(1+x)*arcsin(2*x))/(1-2*x)^(5/2) from the solution of the linear second order differential equation (1-4*x^2)*y''(x) - 2*(8*x+3)*y'(x) - 9*y(x)=0, obtained from the recurrence, with inputs y(0)=0 and y'(0)=1. A special solution is the e.g.f. of the denominators A001879: (1+x)/(1-2*x)^(5/2).

Approximants a(n)/A001879(n) (not in lowest terms): [0/1]; [1/6]; [6/45]; [61/420]; [660/4725]; [8901/62370];..

Approximants in lowest terms: [0/1]; [1/6]; [2/15]; [61/420]; [44/315]; [989/6930]; ...

Sequence in context: A152281 A053462 A160751 this_sequence A034659 A064088 A047737

Adjacent sequences: A142967 A142968 A142969 this_sequence A142971 A142972 A142973

nonn,easy,frac,cofr

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 15 2008