Answer to Question #117131 in Combinatorics | Number Theory for Priya

Question

Find the general form of the solutions of the recurrence relation a_n=8a_(n-2)-16a_(n-4).

Answer 1

characteristic polynomial: x⁴ - 8x²+16 = 0 roots is 2 and -2 => general form of the solutions is

a_n = 2ⁿ * (p*n + q) + (-2)ⁿ * (r*n + s)

if we know the values of a₀ , a₁ , a₂ , a₃ then we get the values p , q , r , s

p = (- 8a₀ - 4a₁ + 2a₂ + a₃) / 32 , q = (16a₀ + 12 a₁ - a₃) / 32 ,

r = (- 8a₀ + 4a₁ + 2a₂ - a₃) / 32 , s = (16a₀ -12a₁ + a₃) / 32