Let f(m,1) = f(1,n) = 1 for m geq 1, n geq 1, and let $f(m,n) = f(m-1,n) + f(m,n-1) +... ?
Let f(m,1) = f(1,n) = 1 for m geq 1 , n geq 1 , and let f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1) for m > 1 and n > 1. Also, let
S(k) = \sum_{a+b=k} f(a,b), \text{ for } a geq 1, b geq 1
Note: The summation notation means to sum over all positive integers a,b such that a+b=k.
Given that
S(k+2) = pS(k+1) + qS(k) \text{ for all } k \geq 2,
for some constants p and q , find pq
Let
Note: The summation notation means to sum over all positive integers
Given that
for some constants
1 Answer
Explanation:
Solving the difference equation
Proposing
we get at
and also
so
and
and thus we have
So solving
we obtain