#puzzle

The integer 72 is the first of three consecutive integers 72, 73, and 74, that can each be expressed as the sum of the squares of two positive integers. The integers 72, 288, and 800 are the first three members of an infinite increasing sequence of integers with the above property. Find a function that generates the sequence and give the next three members.  

We notice that 72 is 6^2 + 6^2 

288 is 12 ^12 

800 is 20^2 + 20^2 

We also note that the next to next number for 72 is merely (6-1)^2 + (6+1)^2 because that will add two 

Similarly the next to next number for 288 is (12-1)^2+(12+1)^2 because that will add two 

And the next to next number for 800 is (20-1)^2+(20+1)^2 

So the middle number must also have a pattern because the function exists. Given that we have three integers to see the pattern, we can generalize it with n 

Consequently the function can be expressed with patterns for all three terms. 

Let us take a closer look at the middle number

73 = 3^2+8^2 = (6-3)^2+(6+2)^2

289 = 8^2 + 15^2 = (12-4)^2+(12+3)^2

801 = 15^2 + 24^2 = (20-5)^2 + (20+4)^2

Therefore given 2n^2 as the first number, the second number is (n-k-2)^2+(n+k+1)^2 and the third number is (n-1)^2 and (n+1)^2 where k is the count of sequence printed starting with 1.
Since the second number is the first number plus 1 we have an equation in n and k
as 2n^2+1 = (n-k-2)^2+(n+k+1)^2
   n = k^2+3k+2
Given k = 4, 5, 6 the next three numbers are 1800, 3528, 6272