Consider N times N as a rectangular array 0 0 0 1 0 2 0 3 1

Consider N times N as a rectangular array: (0, 0) (0, 1) (0, 2) (0, 3) (1, 0) (1, 1) (1, 2) (1, 3) (2, 0) (2, 1) (2, 2) (2, 3) (3, 0) (3, 1) (3, 2) (3, 3) Count them using diagonals as follows: 0 1 3 6 10 2 4 7 11 5 8 12 9 13 14 This defines a function f: N times N rightarrow N where f(m, n) is the number appearing in the (m, n) place. (For example, f(0, 0) = 0, f(3, 1) 13, f(2, 2) = 12.) Provide an explicit formula for f (it is what one calls \"a polynomial function the variables m, n\". It may be a good idea to first find a formula for f(0, n)).

Solution

f(0,n) = sum mation of ( 1 + 2 + .. n) = n(n+1)/2

f(m,0) = summation of (1+ 2 + .. (m+1) } - 1

={( (m+1)(m+2)/2 ) -1}

by inspection we see that

f(m,n) = f(m,o) + f(0,n) + m*n

= {( (m+1)(m+2)/2 ) -1} + {n*(n+1)/2} + m*n