Problem 312

- A Sierpiński graph of order-1 ( S 1 ) is an equilateral triangle.

- S n +1 is obtained from S n by positioning three copies of S n so that every pair of copies has one common corner.

Let C( n ) be the number of cycles that pass exactly once through all the vertices of S n .

For example, C(3) = 8 because eight such cycles can be drawn on S 3 , as shown below:

It can also be verified that :

C(1) = C(2) = 1

C(5) = 71328803586048

C(10 000) mod 108 = 37652224

C(10 000) mod 138 = 617720485



Find C(C(C(10 000))) mod 138.