ClearAll[Padovan1,a,p,s]
p=N[Surd[((9+Sqrt[69])/18),3]+Surd[((9-Sqrt[69])/18),3],200];
s=1.0453567932525329623;
Padovan1[nmax_Integer]:=RecurrenceTable[{a[n+1]==a[n-1]+a[n-2],a[0]==1,a[1]==1,a[2]==1},a,{n,0,nmax-1}]
Padovan2[nmax_Integer]:=With[{},Floor[p^Range[-1,nmax-2]/s+1/2]]
Padovan1[20]
Padovan2[20]
Padovan1[64]===Padovan2[64]
SubstitutionSystem[{"A"->"B","B"->"C","C"->"AB"},"A",10]//Column
(StringLength/@SubstitutionSystem[{"A"->"B","B"->"C","C"->"AB"},"A",31])==Padovan2[32]