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Recursively defined function

Source: Serbia National Olympiad 2016 P2

April 1, 2016
algebracombinatoricsfunctionfloor function

Problem Statement

Let nn be a positive integer. Let ff be a function from nonnegative integers to themselves. Let f(0,i)=f(i,0)=0f (0,i)=f (i,0)=0, f(1,1)=nf (1, 1)=n , and f(i,j)=[f(i−1,j)2]+[f(i,j−1)2] f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] for positive integers i,ji, j such that i∗j>1i*j>1. Find the number of pairs (i,j)(i,j) such that f(i,j)f (i, j) is an odd number.( [x][x] is the floor function).
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