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A358733 Permutation of the nonnegative integers such that A358654(p(n) - 1) = A200714(n) for n > 0 where p(n) is described in Comments.

%I A358733 #51 Apr 21 2024 22:11:04
%S A358733 0,1,2,3,4,5,7,6,8,9,11,10,12,13,17,14,18,15,16,19,20,21,22,27,23,28,
%T A358733 29,24,25,26,30,31,32,33,34,43,35,44,36,37,45,46,47,38,39,40,41,42,49,
%U A358733 48,50,51,52,53,54,55,56,69,57,70,71,58,59,60,72,73,74,75
%N A358733 Permutation of the nonnegative integers such that A358654(p(n) - 1) = A200714(n) for n > 0 where p(n) is described in Comments.
%C A358733 Here p(n) = n + a(d(n)) - d(n) for n > 0 where d(n) = c(b(n)), b(n) = f(g(n) + 2) - n - 1 for n > 0 with b(0) = 0, c(n) = f(g(n) + 3) - n - 1 for n > 0 with c(0) = 0, f(n) = A000045(n) and where g(n) = A072649(n). To compute p(n) we need to know a(d(n)) and to compute a(n) we need to know p(e(n)) where e(n) = n - f(g(n) + 1) for n > 0 with e(0) = 0 in the sense that we can rewrite a(n) = n + [e(n) > 0]*(a(h(n)) - h(n) ...) (here h(n) = d(e(n))) as a(n) = n - e(n) + [e(n) > 0]*(p(e(n)) ...).
%H A358733 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the nonnegative integers</a>
%F A358733 a(n) = n + [e(n) > 0]*(a(h(n)) - h(n) - f(s(n)) + [s(n) mod 2 = g(n) mod 2]*f(g(n) - 2)) for n > 0 with a(0) = 0 where s(n) = g(e(n) - 1) (here we also consider that g(0) = 0), h(n) = d(e(n)), e(n) = n - f(g(n) + 1) for n > 0 with e(0) = 0, d(n) = c(b(n)), b(n) = f(g(n) + 2) - n - 1 for n > 0 with b(0) = 0, c(n) = f(g(n) + 3) - n - 1 for n > 0 with c(0) = 0, f(n) = A000045(n) and where g(n) = A072649(n).
%o A358733 (PARI) g(n)=local(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2) \\ from A072649
%o A358733 d(n) = { while(n>0, my(A=g(n), B=fibonacci(A)); n-=B; if(B>n, break)); n; }
%o A358733 a(n) = if(n>0, my(A=g(n), B=fibonacci(A+1), C=n-B, D=d(C), E=g(C-1)); n + if(C>0, a(D) - D - fibonacci(E) + if(E%2==A%2, fibonacci(A-2))))
%Y A358733 Cf. A000045, A048679, A072649, A200648, A200714, A348366, A358654.
%K A358733 nonn,base
%O A358733 0,3
%A A358733 _Mikhail Kurkov_, Mar 13 2023 [verification needed]