Select primes from their Zeckendorf representation

Question

I'm working with the Zeckendorf representation of prime numbers. I'm using

ResourceFunction["ZeckendorfRepresentation"][Prime[n]]

and I would like to select from all the results, the ones with this peculiar representation: an alternating sequence of $ 1 $ and $ 0 $ of various lenght, ending with two adjacent $ 0 $ and $ 1 $ (and eventually just various $ 0 $). As an example I want to select $ 607 $ because this is the representation:

$ \{1,0,1,0,1,0,1,0,1,0,0,1,0\} $

or $ 6763 $:

$ \{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1\} $

Or $ 679891411787178541 $

$\{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\}$

Answer 1

Something along these lines?

thePattern = {PatternSequence[1, 0] .., 0, 1, 0 ...};
MatchQ[ResourceFunction["ZeckendorfRepresentation"][607], thePattern]

True

MatchQ[ResourceFunction["ZeckendorfRepresentation"][6763], thePattern]

True

MatchQ[ResourceFunction["ZeckendorfRepresentation"][679891411787178541], thePattern]

True

UPDATE

If you want the primes themselves that satisfy this property:

primes = Array[Prime, 1000];
Select[primes, MatchQ[thePattern]@*ResourceFunction["ZeckendorfRepresentation"]]

{19, 31, 53, 131, 139, 607, 953, 1453, 2351, 2579, 6763}

If you want the zeckendorf representations:

primeZecks = Array[ResourceFunction["ZeckendorfRepresentation"]@*Prime, 1000];
Select[zeckendorfs, MatchQ[thePattern]]

Or, if you'll want both forms, then you could create pairs, or if you want ... etc etc etc