Arithmetic Progressions in the Values of Quadratic Polynomials and Some Figurate Numbers.

In this paper, we investigate arithmetic progressions in integer-valued quadratic polynomials. We prove two results for integer-valued quadratic polynomials. First, we prove that there is no three-term arithmetic progression with a perfect square as a common difference for a class of integer-valued quadratic polynomials. We then find a class of numbers depending on integer-valued quadratic polynomials, which cannot work as a common difference for three-term arithmetic progression. Further, we deduce some results for arithmetic progressions in the sequence of polygonal and centered polygonal numbers. [ABSTRACT FROM AUTHOR]