Based on Ghol's theory that this usage of reckon might be Australian I checked the Macquarie Dictionary but it only lists: reckon on, reckon up, reckon with, reckon without.
The OED2 on the other hand does have an entry for it:
2.a. To count, so as to ascertain the number or amount of; to ascertain or arrive at (a number, quantity, etc) by counting or calculating; to compute. Also with out.
1633 T. Adams Exp. 2 Peter iii 10.1307 A woman reckons out her nine moneths, and can guesse neare to the day of her comming.
Based on the single example given by the OED2, "reckon out" might be older usage and possibly rare or dialectal these days. So the LEO entry would appear to be valid but should possibly be marked [rare]. Using Google I was able to find a few instances of "reckon out" likely to have been written by native speakers of English. Examples:http://www.ku.edu/carrie/archives/wwi-l/2000/...
For Americans - check out the increased US "liquidity" - the amount of gold and silver bullion in the US - after the Great War and see if you can reckon out
where it came from if not from Russian payments to reverse the Peace of Brest-Litovsk.http://www.geocities.com/gospelandrevival/mes...
We find the Lord Jesus turning to thousands of followers in Lk.14:25-35 and telling them just this: go back and reckon out first what the price of following Me is.http://www.nzetc.org/etexts/JCB-014.pdf
I reckon out £2 for living in all its details & £1 for pleasures or rather education in a broad sense, books, music plays etc.http://members.aol.com/steveiron/quotes.html
The most skillful chess-player, the cleverest of them, can only look a few moves ahead; a French player who could reckon out ten moves ahead was written about as a marvel.