This problem is related to the so-called Diophantine representation of a
sequence of integers, and for some results we refer to the papers of M.J.
DeLeon [3], V.E. Hoggatt, M. Bicknell-Johnson [6], J.P. Jones [7], [8], [9],
and W.L. McDaniel [11]. Also, it is connected to the Y.V. Matiyasevich and
J. Robertson way to solve the Hilbert's Tenth Problem, and it has applications to the problem of singlefold Diophantine representation of recursively enumerable
sets. In the recent paper of R. Keskin, N. Demiturk [10] the equations
x2