HI Cphill

I did not really understand exactly second part of question but I have interpreted it as follows:

Define Pattern :

S_{c = 13 + }_{33 + 53..........+ (2n-1)3 }_{ = n2(2n2-1) (as per formula for sum of cubes for odd integers)}

Now Sum of cubes for n+(n+1)

S_{2}_{n+1 = n2(2n2-1) + [ 2(n+1)-1]3}

= 2n^{4}- n^{2} + [2n+1]^{3}

= 2n^{4}- n^{2} + 8n^{3}+ 12n^{2 }+ 6n + 1

= (2n^{4} + 4n^{3} + 2n^{2}) + (4n^{3} + 8n^{2} + 4n) + (n^{2} + 2n + 1)

= 2n^{2}(n+1)^{2 + }4n(n+1)^{2} + (n+1)^{2}

^{= }(n+1)^{2}(2n^{2} + 4n + 1)