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A319952 Let M = A022342(n) be the n-th number whose Zeckendorf representation is even; then a(n) = A129761(M).

%I A319952 #15 Oct 03 2018 15:27:34
%S A319952 1,2,3,1,6,1,2,11,1,2,3,1,22,1,2,3,1,6,1,2,43,1,2,3,1,6,1,2,11,1,2,3,
%T A319952 1,86,1,2,3,1,6,1,2,11,1,2,3,1,22,1,2,3,1,6,1,2,171,1,2,3,1,6,1,2,11,
%U A319952 1,2,3,1,22,1,2,3,1,6,1
%N A319952 Let M = A022342(n) be the n-th number whose Zeckendorf representation is even; then a(n) = A129761(M).
%C A319952 The Zeckendorf representations of numbers are given in A014417. The even ones are specified by A022342.
%C A319952 The offset here is 2 (because A129761 should really have had offset 1 not 0).
%F A319952 If the Zeckendorf representation of M ends with exactly k zeros, ...10^k, then a(n) = ceiling(2^k/3).
%p A319952 with(combinat): F:=fibonacci:
%p A319952 A130234 := proc(n)
%p A319952     local i;
%p A319952     for i from 0 do
%p A319952         if F(i) >= n then
%p A319952             return i;
%p A319952         end if;
%p A319952     end do:
%p A319952 end proc:
%p A319952 A014417 := proc(n)
%p A319952     local nshi, Z, i ;
%p A319952     if n <= 1 then
%p A319952         return n;
%p A319952     end if;
%p A319952     nshi := n ;
%p A319952     Z := [] ;
%p A319952     for i from A130234(n) to 2 by -1 do
%p A319952         if nshi >= F(i) and nshi > 0 then
%p A319952             Z := [1, op(Z)] ;
%p A319952             nshi := nshi-F(i) ;
%p A319952         else
%p A319952             Z := [0, op(Z)] ;
%p A319952         end if;
%p A319952     end do:
%p A319952     add( op(i, Z)*10^(i-1), i=1..nops(Z)) ;
%p A319952 end proc:
%p A319952 A072649:= proc(n) local j; global F; for j from ilog[(1+sqrt(5))/2](n)
%p A319952        while F(j+1)<=n do od; (j-1); end proc:
%p A319952 A003714 := proc(n) global F; option remember; if(n < 3) then RETURN(n); else RETURN((2^(A072649(n)-1))+A003714(n-F(1+A072649(n)))); fi; end proc:
%p A319952 A129761 := n -> A003714(n+1)-A003714(n):
%p A319952 a:=[];
%p A319952 for n from 1 to 120 do
%p A319952    if (A014417(n) mod 2) = 0 then a:=[op(a), A129761(n-1)]; fi;
%p A319952 od;
%p A319952 a;
%Y A319952 Cf. A003714, A014417, A022342, A072649, A129761, A130234.
%K A319952 nonn
%O A319952 2,2
%A A319952 _Jeffrey Shallit_ and _N. J. A. Sloane_, Oct 03 2018