Recurrence Relation

Algebra Level 5

Brilli the ant has thought up a diabolical sequence of integers \(a_n\). It has initial values \(a_1 = 1 \) and \( a_2 = 3 \). Subsequent terms are given by

\[ a_n = (n+1) a_{n-1} - n a_{n-2} \quad \mbox{ for } n\geq 3.\] Brilli the ant wants to know, how many integer values of \(n\) from 1 to 1000 (inclusive) are there such that \(a_n\) is a multiple of 11?