## Consecutive die throws

### Problem 423

Published on Sunday, 14th April 2013, 10:00 am; Solved by 212Let `n` be a positive integer.

A 6-sided die is thrown `n` times. Let `c` be the number of pairs of consecutive throws that give the same value.

For example, if `n` = 7 and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:

(__1,1__,5,6,6,6,3)

(1,1,5,__6,6__,6,3)

(1,1,5,6,__6,6__,3)

Therefore, `c` = 3 for (1,1,5,6,6,6,3).

Define C(`n`) as the number of outcomes of throwing a 6-sided die `n` times such that `c` does not exceed π(`n`).^{1}

For example, C(3) = 216, C(4) = 1290, C(11) = 361912500 and C(24) = 4727547363281250000.

Define S(`L`) as ∑ C(`n`) for 1 ≤ `n` ≤ `L`.

For example, S(50) mod 1 000 000 007 = 832833871.

Find S(50 000 000) mod 1 000 000 007.

^{1} π denotes the **prime-counting function**, i.e. π(`n`) is the number of primes ≤ `n`.