Basic Examples (2) 

Find the binomial number representation for the first twenty integers:

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Find the binomial number representation for a larger non-negative integer:

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Verify that the following sum gives back the original input:

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Scope (1) 

By determining a tight bound for each item in the triplet before initiating the solution search, the internal algorithm allows the function to process large inputs efficiently:

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Properties and Relations (4) 

There is a unique representation for zero:

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It is based on the fact that Binomial[n,k] vanishes for integers n and k such that n<k:

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For those input values exactly equal to Binomial[k,3], the returned triplets always begin with a=0 and b=1:

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For integers from 1 to n, the subsequence formed by numbers with a=1 in the binomial number representation is OEIS A126862:

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The ResourceFunction SubsetFromIndex uses a similar algorithm:

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Possible Issues (2) 

The input must be a non-negative integer. Otherwise the function returns unevaluated:

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The uniqueness of the representation is lost if {a,b,c} is not strictly increasing:

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Imposing the order ensures that the solution is unique. For instance:

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Neat Examples (2) 

Visualize the "digit sum" of the binomial number representation for some non-negative integers:

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A graphic for the progression of triples:

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