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Cyclic paths on Sierpiński graphs

Problem 312

Published on Sunday, 28th November 2010, 01:00 am; Solved by 515; Difficulty rating: 50%

- A Sierpiński graph of order-1 (S1) is an equilateral triangle.
- Sn+1 is obtained from Sn by positioning three copies of Sn so that every pair of copies has one common corner.

p312_sierpinskyAt.gif

Let C(n) be the number of cycles that pass exactly once through all the vertices of Sn.
For example, C(3) = 8 because eight such cycles can be drawn on S3, as shown below:

p312_sierpinsky8t.gif

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.