Puzzle #187: Exact Change

Squareland has coins in several denominations. We will concern ourselves with only the 4-, 9-, 16-, 25-, 36-, 49-bit coins. They used to have a 1-bit coin but no more. You want to give someone exactly 50 bits in coin. How many ways are there to do this?

Hint: You could just start iterating through the possibilities, but there is a shortcut that will make this much easier.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, January 4, 2017 at noon Pacific Time. I will post the answer shortly after.