MathDB

Let

n

be a positive integer.

f(n)

denotes the number of

n

-digit numbers

\overline{a_1a_2\cdots a_n}

(wave numbers) satisfying the following conditions : (i) for each

a_i \in\{1,2,3,4\}

a_i \not= a_{i+1}

i=1,2,\cdots

; (ii) for

n\ge 3

(a_i-a_{i+1})(a_{i+1}-a_{i+2})

is negative,

i=1,2,\cdots

. (1) Find the value of

f(10)

; (2) Determine the remainder of

f(2008)

upon division by

13

Problem Statement