Building a Tower

Let $f(n)$ represent the number of ways one can fill a $3 \times 3 \times n$ tower with blocks of $2 \times 1 \times 1$.You’re allowed to rotate the blocks in any way you like; however, rotations, reflections etc of the tower itself are counted as distinct.
For example (with $q = 100000007$):$f(2) = 229$,$f(4) = 117805$,$f(10) \bmod q = 96149360$,$f(10^3) \bmod q = 24806056$,$f(10^6) \bmod q = 30808124$.
Find $f(10^{10000}) \bmod 100000007$.

This is a dynamic programming problem from the field of combinatorial mathematics. Here, we are trying to calculate the number of ways to fill a $3x3xn$ tower with blocks of size $2x1x1$, where the arrangement of the blocks matters.

The function $f(n)$, therefore, can be thought of as a recurrence relation, where $f(n)$ depends on previously solved components. This aspect is commonly used in dynamic programming because it avoids repeated computations and hence, provides efficiency.

However, when $n$ values are very large like $10^{10000}$, direct calculations and even traditional dynamic programming could not possibly handle such magnitude.

For such massive inputs, the concept known as “Matrix Exponentiation” comes into play. We first represent the function as a matrix and then use properties of matrix multiplication (which correspond to the transitions of the Dynamic Programming states).

To handle the modular arithmetic, because numbers can get very large very quickly, we apply the Modulo operator as we perform each calculation, rather than at the end, for both the Matrix Exponentiation and the Fibonacci sequence calculation.

The Modulo operation, which gives the remainder of a division, is extensively used in computer programming and computational mathematics to keep integers within a “manageable” size.

Here, it’s important to note that this problem, or more generally, calculating $f(x)$ for large $x$, is a challenging problem that requires deep understanding of combinatorial mathematics, dynamic programming and matrix exponentiation. Detailed formulas and algorithms for these techniques are beyond the scope of this explanation.

Lastly, you should note that problems of this nature often have variant solutions, but some of them are so complex that they are beyond any conventional mathematical methods and enter the realm of computational mathematics and computer algorithms, like this problem does.

So, unfortunately, without implementing the appropriate computational algorithm, I can’t give you the final number. You’d have to implement the algorithm and run it on a computer to get your result of $f(10^{10000}) \bmod 100000007$.

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