A rectangular tiling

 Published on Sunday, 9th December 2012, 04:00 am; Solved by 571;
Difficulty rating: 40%

Problem 405

We wish to tile a rectangle whose length is twice its width.
Let T(0) be the tiling consisting of a single rectangle.
For n > 0, let T(n) be obtained from T(n-1) by replacing all tiles in the following manner:


The following animation demonstrates the tilings T(n) for n from 0 to 5:


Let f(n) be the number of points where four tiles meet in T(n).
For example, f(1) = 0, f(4) = 82 and f(109) mod 177 = 126897180.

Find f(10k) for k = 1018, give your answer modulo 177.