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Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=2 (see A130249 for another version). a(n+1) is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).
a(n)=ceiling(log_2(3n-1))=1+floor(log_2(3n-2)) for n>=1. Also true: a(n)=A130249(n-1)+1=A130253(n-1) for n>=1. G.f.: g(x)=x/(1-x)*sum{k>=0, x^A001045(k)}.