Freeflows

Recursive Function Sequence

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 $.

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$\frac{a}{b}$
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$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
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$\tan{}$

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$)$
$[$
$]$
$\cap$
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$\infty$

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