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 One possible answer to what's next by Dej Mar)

Hi Dej!

You might like to try looking for patterns in the right-hand diagonals i.e. the ones parallel to the series of 1's. To get you started, the next row is actually: 2048n^6+5632n^5+5760n^4+2688n^3+560n^2+42n+1

The last element in each row is 1; the penultimate one is A002378 in Sloane and the sum of the coefficients is A011900 in Sloane.

That should be enough of a hint for now, I think...

Edited on June 27, 2011, 9:04 am

Posted by broll on 2011-06-27 08:54:36