A rectangular tiling

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

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter