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### Keeping memory to reduce the running time of recursion [duplicate]

I am using the following recursion in Mathematica to compute W[n, 1, s] for given n and s: ...
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### Can one identify the design patterns of Mathematica?

... or are they unnecessary in such a high-level language? I've been thinking about programming style, coding standards and the like quite a bit lately, the result of my current work on a mixed ...
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### Performance tuning in Mathematica?

What performance tuning tricks do you use to make a Mathematica application faster? MATLAB has an amazing profiler, but from what I can tell, Mathematica has no similar functionality.
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### Why does Mathematica use [[ ]] notation for array indexing?

I am confused by why Mathematica uses [[3]] to get the 3rd element, or [[i,j] to get the i,j-th element of a 2D array. This ...
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### Compiling a recursive formula

My question is related to computing what is called "invariant measure" for a particular well known fractal - the Sierpinski triangle. We have an array m of four two by two matrices, say ...