Re: Small Sunday Afternoon MiniChallenge Message #17 Posted by Gerson W. Barbosa on 8 May 2010, 11:02 a.m., in response to message #16 by Paul Dale
Hi Paul,
You have noticed
16384 * 13/8 * 3465/2048 = 45045
131072 * 15/8 * 45045/16384 = 675675
So the next term will be
1048576 * 17/8 * 675675/131072 = 11486475
which of course is correct. Hadn't seen it.
I've found a way to express this sequence in terms of the Gamma Function:
a(n)=(2^(2+n)*Gamma(n+5/2))/(3*sqrt(pi))
Thus
a(6)=(2^(2+6)*Gamma(6+5/2))/(3*sqrt(pi)) = 675675
a(7)=(2^(2+7)*Gamma(7+5/2))/(3*sqrt(pi)) = 11486475
However, your insight is right: this can be expressed in terms of the double factorial function and the equivalent expression is more elegant and simple.
Part b is more a guessing game rather than a thinking game. If
1 * 2 * 3 * 4 * 5 * 6 ... * n produces special values at 1.5, 2.5, 3.5, 4.5 ... all involving sqrt(pi)
and
1 * 1.5 * 2 * 2.5 * 3 * 3.5 * ... * n produces special values at 1.25, 1.75, 2.25, 3.75, ... all of them involving sqrt(pi/2)
then you can guess what function will produce special values involving sqrt(pi/4) at 1.125, 1,375, 1,625, 1,875, 2,125 ...
Gerson.
