Sequence of numbers
Source: Vietnam NMO 1995, Problem 2
September 7, 2008
number theory proposednumber theory
Problem Statement
The sequence (a_n) is defined as follows:
a_0\equal{}1, a_1\equal{}3
For , a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}9a_n if n is even, a_{n\plus{}2}\equal{}9a_{n\plus{}1}\plus{}5a_n if n is odd.
Prove that
1) (a_{1995})^2\plus{}(a_{1996})^2\plus{}...\plus{}(a_{2000})^2 is divisible by 20
2) a_{2n\plus{}1} is not a perfect square for every natural numbers .