MathDB

Let

a_0, a_1, a_2, \dots

be an infinite sequence of positive integers with the following properties: -

a_0

is a given positive integer; - For each integer

n \geq 1

a_n

is the smallest integer greater than

a_{n-1}

such that

a_n + a_{n-1}

is a perfect square. For example, if

a_0 = 3

, then

a_1 = 6

a_2 = 10

a_3 = 15

, and so on.
(a) Let

T

be the set of numbers of the form

a_k - a_l

, with

k \geq l \geq 0

integers. Prove that, regardless of the value of

a_0

, the number of positive integers not in

T

is finite.
(b) Calculate, as a function of

a_0

, the number of positive integers that are not in

T

Problem Statement