OFFSET
0,4
COMMENTS
a(n) = s(1)s(2)...s(n)(1/s(2) - 1/s(3) + ... + c/s(n)) where c=(-1)^n and s(k) = 2k-1 for k = 1,2,3,...
REFERENCES
A. E. Jolliffe, Continued Fractions, in Encyclopaedia Britannica, 11th ed., pp. 30-33; see p. 31.
FORMULA
A024199(n)/a(n) -> Pi/(4-Pi) as n -> oo. - Max Alekseyev, Sep 23 2007
E.g.f.: (1-Pi/4)/sqrt(1-2*x) + 1/2*log(2*x+sqrt(4*x^2-1))/sqrt(2*x-1). - Vaclav Kotesovec, Mar 18 2014
a(n) ~ (4-Pi) * 2^(n-3/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 18 2014
MATHEMATICA
CoefficientList[Series[(1-Pi/4)/Sqrt[1-2*x] + 1/2*Log[2*x+Sqrt[4*x^2-1]]/Sqrt[2*x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 18 2014 *)
RecurrenceTable[{a[n+1] == 2*a[n] + (2*n-1)^2*a[n-1], a[0] == 1, a[1] == 0}, a, {n, 0, 20}] (* Vaclav Kotesovec, Mar 18 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Revised by N. J. A. Sloane, Jul 19 2002
Initial terms changed by Max Alekseyev, Sep 23 2007
STATUS
approved