how can we prove that an=(an/2)+n, a1=0 where n is a power of 2 is a linear recurrence relation. (Note n is a subscript)

How can we prove that the generating function of the sequence {1, 2, 3, 4, …, n …} is (1–z)^(-2) (Note ^ symbol represents power)

How can i prove that Any graph is isomorphic to its complement

How to prove that Any bipartite graph is planer.

Raghu goes to a grocery shop and purchases grocery for Rs. 23. He has 3 five rupee coins, 4 two rupee coins and 6 one rupee coins. In how many ways can he pay the shop keeper? Find a solution using generating functions

Consider the number of words of length n made by using the letters ‘a’ and ‘b’
and which do not contain 2 consecutive ‘a’s. Denote this number by an.
i) What are the values of a1, a2, a3 and a4?
ii) Derive a recurrence relation for an, and solve it.