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→Values for discrete regular lattices: Add closed form expression for the surfae function. |
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:<math> S_{d+1}(r) = 2+S_{d}(r)+2\sum^{r-1}_{i=1}S_{d}(i). </math>
From combinatorics the surface function for a regular lattice can be written as
:<math> S_{d}(r) = \sum^{d-1}_{i=0}(-1)^{i}2^{d-i}{d \choose i} { d+r-i-1 \choose d-i-1 }. </math>
The following expression for the sum of positive integers raised to a given power <math>\textstyle k</math> will be useful to calculate the surface function for higher values of <math>\textstyle d</math>:
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