Combinatorics | Number Theory Answers

Solve the recurrence relation a_n=-3a_(n-1)-3a_(n-2)-a_(n-3) with a_0=5,a_1=-9,and a_2=15.

Solve the recurrence relationa_n=6a_(n-1)-12a_(n-2)+8a_(n-3) with a_0=-5,a_1=4,and a_2=88.

Find an explicit formula for the Fibonacci numbers.

How many bit sequences of length seven contain an even number of 0s?

Find a recurrence relation for the number of bit sequences of length n with an even number of 0s.

Solve a_n=6a_(n-1)-9a_(n-2)+F(n) when
(i)F(n)=3n
(ii) F(n)=n3^n
(iii) F(n)=n^2 2^n
(iv)F(n)=〖(n〗^2+1).3^n

Show that the sequence 〖{a〗_n} is a solution of the recurrence relation a_n=a_(n-1)+〖2a〗_(n-2)+2n-9 if
(i)〖 a〗_n=-n+2
(ii) a_n=〖5(-1)〗^n-n+2
(iii) 〖3(-1)〗^n+2^n-n+2
(iv)〖7.2〗^n-n+2

Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. Find also the initial conditions?

Is the sequence 〖{a〗_n} a solution of the recurrence relation a_n=8a_(n-1)-16a_(n-2)if a_n=n^4 a_n=0.

In a company, ID cards have 5 digit numbers.

a) How many ID cards can be formed if repetion of the digit is allowed?

b) How many ID cards can be formed if repetion of the digit is not allowed?