2010 Fermat Problem 24

Spheres can be stacked to form a tetrahedron by using triangular layers of spheres. Each sphere touches the three spheres below it. The diagrams show a tetrahedron with four layers and the layers of such a tetrahedron. An *internal sphere* in the tetrahedron is a sphere that touches exactly three spheres in the layer above. For example, there is one internal sphere in the fourth layer, but no internal spheres in the first three layers.

A tetrahedron of spheres is formed with thirteen layers and each sphere has a number written on it. The top sphere has a 1 written on it and each of the other spheres has written on it the number equal to the sum of the numbers on the spheres in the layer above with which it is in contact. For the whole thirteen layer tetrahedron, the sum of the numbers on all of the internal spheres is

\textbf{(A)}\ 772\,588\quad \textbf{(B)}\ 772\,566\quad \textbf{(C)}\ 772\,156\quad \textbf{(D)}\ 772\,538\quad \textbf{(E)}\ 772\,626

If there are no answer choices shown, enter a numerical answer.

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