Think of an infintie set with a perfect square and/or perfect cube that can' t be found by inspection (big #)

Question

I'm doing research on perfect powers of infintie sets and so far all the sets i've used:

{11, 111, 1111...}
{45, 454, 4545, 45454 ...}
{83, 838, 8383, 83838 ...}
don't have perfect square
(didn't do the cube yet)

I need some sort of infinite set (with a pattern like mines) that does have a perfect square/cube that can't be found by inspection, for ex. it is like a huge number or something..

Thanks

Answer 1

Ok, here's an example that you might like...

Define the sequence s[n] as follows:
s[1] = 1
s[2] = 3
s[3] = 5
for all n > 3, s[n] = s[n-3] + s[n-2] + s[n-1]

So, the sequence goes like this:
1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, 2209, 4063,
7473, 13745, 25281, 46499, 85525, 157305, 289329,
532159, 978793, 1800281, 3311233, ....

There are some perfect squares in the sequence.
s[4] = 9 = 3 * 3
s[13] = 2209 = 47 * 47
s[17] = 25281 = 159 * 159

s[13] and s[17] can't be easily found by inspection.