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Consider a random number pyramide (see e.g. http://www.puzzle-magazine.com/numbertower.jpg).

Assume that the numbers in the bottom line are equidistantly distributed. Let $s$ denote the left-most number in the bottom line and let $d$ denote the distance between the numbers in the bottom line.

By experiment I have then found that the number $N$ in the $m^{\text{th}}$ box (counted from left) in the $n^{\text{th}}$ row (counted from the bottom) is given by

\begin{equation} N(m,n) = (2s + d(2m + n - 3)) \cdot 2^{n - 2}. \end{equation}

How do I prove that this is correct (if it indeed is correct)?

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    Maybe by induction. The base case would be that the bottom row $n=1$ satisfies indeed the formula. Then you would need to show that the $n+1$ row also satisfies the formula assuming it is true for $n$. and using the fact that, by definition of a "pyramid" we must have $N(m,n+1) = N(m,n) + N(m+1,n)$. More precisely : Using [this method](http://math.stackexchange.com/questions/7660/induction-on-two-integer-variables) – 2017-02-16
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    @Zubzub That was the way to do it. Thank you for the hint. – 2017-02-16

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