It is P(k+2), of course: if k is an odd positive integer, the next ODD integer is k + 2, not k + 1.

Without induction: n^2 - 1 = (n-)(n+1). As n is odd, exactly one of n-1 or n+1 is divisible by 4 and the other factor is even ==> the whole thing is divisible by 4*2 = 8.

Tonio