[seqfan] Re: 1, 18, 450, 12420, 360450... 3*sqrt(3)/pi.

[Brad Klee]

Hi Seqfans,

Some time ago I was having trouble finding the area of
a regular hexagon (haha) [1]. No one, not even Andrew
Hone, said the following:

The period function

T(x)= 1 + (1/2)*x + (25/72)*x^2 + (115/432)*x^3 + ...

satisfies an ordinary differential equation:

120*T(x)
+ 5*(417*x - 368 )*T^1(x)
+ (3575*x^2 - 6041*x + 2394)*T^2(x)
+ 36*(x-1)*(40*x^2 - 59*x + 15)*T^3(x)
+ (36*x*(4*x-5)*(x-1)^2)*T^4(x) = 0 .

One interesting fact about this T(x) is that:

Int_{0..1} dx*T(x) = 3*sqrt(3)/pi .

Unfortunately, this one fact probably doesn't justify
inclusion of the corresponding integer sequence to
OEIS.

Cheers,

Brad

[1] http://list.seqfan.eu/pipermail/seqfan/2017-July/017807.html