A sequence {B_p} of positive integers is such that:

B₁ = 20, B₂ = 30, and:

B_p+2 = 3*B_p+1 – B_p, whenever p ≥ 1.

Determine all possible positive integer values of p such that:
1 + 5*B_p+1*B_p is a perfect square.

**** For an extra challenge, derive a non computer program assisted solution.