Each of A,B and C is a positive integer such that each of the expressions A*B+A+B, B*C+B+C, C*A+C+A, A*B+C and A*C+B is a perfect square.

(I) Find the smallest value of A+B+C.

(II) Does there exist an infinity of triplets (A,B,C) satisfying the given conditions?

Give reasons for your answer.

(In reply to

Two small solutions and a conjecture by Brian Smith)

After some more exploration I have come up with this conjecture:

Any consecutive three terms of OEIS A027941 (0, 1, 4, 12, 33, 88, 232, ...) form a triplet ABC which satisfies the given problem.

Let F(n) be the nth term of the Fibonacci sequence. Then for parameter k:

A = F(2k-1)-1

B = F(2k+1)-1

C = F(2k+3)-1

A*B+A+B = F(2k)^2

A*C+A+C = F(2k+1)^2

B*C+B+C = F(2k+2)^2

A*B+C = [F(2k)+1]^2

A*C+B = [F(2k+1)-1]^2

B*C+A = [F(2k+2)-1]^2