Modified Zeckendorf represention

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Given x as searched integer and k as minimum zeros between 2 ones.

Example:

x := 7,k := 0 Binary represention: 111

x := 7,k := 1 Binary represention: 1010

x := 7,k := ?(>1) Binary represention: ?

How can i calculate the binary represention for k > 1? I know that for k := 0 I use the binary represention, for k := 1 i use the zeckendorf represention (a 1 represented by a fibonacci number)

But how can i create the bit string for k > 1?

Thx in advance.

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