Find a closedform formula for this following linear homogene

Find a closed-form formula for this following linear homogeneous recurrence relation with constant coefficients. Do not round off or use calculator approximations: use exact arithmetic! a_0 = -4, a_1 = -3, a_2 = 0, and a_n = -3a_n-1 - 3a_n-2 - a_n-3, n ge 3 A library has four identical display cases that are used to promote new acquisitions. This month, the librarians wish to promote nine books. They do not want any empty display cases. In how many ways can the books be displayed?

Solution

13) a3 = -3a2-3a1-a0 = 21

a4 = -3a3-3a2-a1 = -63+9 =-54

In general

an = -3a_n-1-3a_n-2-a_n-3

= -3 (-3a_n-2-3a_n-3-a_n-4)-3a_n-2-a_n-3

= 6a_n-2+8a_n-3+3a_n-4

= 6(-3a_n-3-3a_n-4-a_n-5)+8a_n-3+3a_n-4

= -10a_n-3-15a_n-4+6a_n-5

This trend continues till we go to a0,a1,a2.

14) No of display cases = 4

No of books to be promoted = 9

No case should be empty

Hence first fill up the 4 displays with 4 books.

This can be done in one way

REmaining books = 5

Each book can be placed in any one of the 4 identical cases

Hence no of ways = 1x5⁴ = 625 ways