The fourth power (Juzuk)

J, 33

~.+/\(#~i.200)(+/*2|#)/.1+i.2!200

J, 40 38 (not summing unneeded)

+/\+/"1(+:i.100){(#~i.201)[/.>:i.20100

J, 30 (summing unneeded)

+/\_2{.\(#~i.201)+//.>:i.20100

Haskell, 78

print[sum$concat$take x[take x$drop(sum[1..x-1])[1..]|x<-[1,3..]]|x<-[0..100]]
[0,1,16,81,256,625,1296,2401,4096,6561,10000,14641,20736,28561,38416,50625,65536,...

Hope the slightly different output formatting is ok. There's probably a much better way to write this in Haskell, but I felt like solving this in a language I don't often use.