# Problem 423: Consecutive die throws
Let 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.