Given that n is a whole number, what is the next expression in this series of expressions, each of which is valid for all n?

n^0,
2n+1,
8n^2+6n+1,
32n^3+40n^2+12n+1

(In reply to Hint 2 by broll)

 2048n^6+5632n^5+5760n^4+3584n^3+600n^2+42n+1

Where x is the xth term of the sequence, and where the intermediate calculations are non-zero integers:
The first term of the polynomial appears to follow the pattern
[2^(2x+1)*n^x].
The second term of the polynomial appears to follow the pattern
[2^(2x-1)*(2x+1)*n^(x-1)].
The third term of the polynomial appears to follow the pattern
[2^(2x-1)*(2x+1)*(x)*n^(x-1)].
The coefficient of each n term of the polynomial appears to follow the pattern  [x*(x+1)].
The other patterns remain obscure, I selected one to post here from the several possible.

Posted by Dej Mar on 2011-06-27 06:41:02