444444....4444888888....8889

[Where there are 'k' Fours, '(k-1)' Eights and 'Exactly One' 9],

are always perfect squares.

(For example the sequence of numbers: 49, 4489, 444889, ....etc. and so on are always perfect squares).

The number 44444..48888..89 can be written as:
N=1+4444...4888..8=1+8(1+10+...+10^(k-1))+4*10^k*(1+10+...+10^(k-1))
N=1+(4*10^k+8)*((1+10+...+10^(k-1))
N=1+4*(10^k+2)*(10^k-1)/9     (Sum of k terms of GP)
N=1+(4/9)*(10^(2k)-2+10^k)
N=1/9*(9+4*10^2k-8+4*10^k)
N=1/9*(1+4*10^2k+4*10^k)
N=((2*10^k+1)/3)²
So, N is a perfect square.