strange pattern in 3 squares in arithmetic progression

[tmori]

Hi,

I was working on a problem that involved 3 squares in arithmetic progression when I came upon this strange pattern that I can't explain...

if we let a, b and c equal 1, 5 and 7 respectively then we have a^2, b^2, c^2 as an arithmetic progression of 3 squares 1, 25, 49

Keeping a as 1 and checking all possible values of b up to 20,000,000 we get the following values for a, b and c that produce 3 squares in arithmetic progression...

a 1 1 1 1 1 1 1 1 1

b 5 29 169 985 5741 33461 195025 1136689 6625109

c 7 41 239 1393 8119 47321 275807 1607521 9369319

Taking the above values of b we find that

29 = 6 * 5 - 1
169 = 6 * 29 - 5
985 = 6 * 169 - 29
5741 = 6 * 985 - 169
33461 = 6 * 5741 - 985
195025 = 6 * 33461 - 5741
1136689 = 6 * 195025 - 33461
6625109 = 6 * 1136689 - 195025

i.e. each successive value of b can be calculated from previous ones. Am I missing something blindingly obvious? (btw, it is the same for the values of c).

Thanks

 

[tmori]

Thank you Dennis for your non-response.

Yes I am aware of the Sloane series' you linked to and am also aware of the intricate link between 3 squares in arithmetic progression and Pythagorean triples (and Pell etc.).

As I am on limited time at this moment and wish to delve much deeper into this and, hoping that someone might give the problem a bit more thought than merely googling a series of numbers, i will pose a deeper question regarding this tomorrow...