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A264596 Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.

%I A264596 #40 Nov 05 2023 09:04:04
%S A264596 0,1,1,3,1,4,3,7,1,6,4,10,3,10,7,15,1,10,6,16,4,15,10,22,3,16,10,24,7,
%T A264596 22,15,31,1,18,10,28,6,25,16,36,4,25,15,37,10,33,22,46,3,28,16,42,10,
%U A264596 37,24,52,7,36,22,52,15,46,31,63,1,34,18,52,10,45,28
%N A264596 Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.
%H A264596 Alois P. Heinz, <a href="/A264596/b264596.txt">Table of n, a(n) for n = 0..20000</a>
%F A264596 a(2^n) = 1.
%F A264596 a(2^n-1) = 2^n-1.
%F A264596 a(2n) = a(n), a(2n+1) = a(n) + n+1, a(0) = 0. - _Alois P. Heinz_, Nov 19 2015
%F A264596 Conjecture: a(n) = n*(n + 3)/2 - A007814(A293290(n)) for n > 0. - _Velin Yanev_, Sep 12 2017
%e A264596 S_0 = [0], a(0) = 0;
%e A264596 S_1 = [0, 1], a(1) = 1;
%e A264596 S_2 = [0, 01, 1], a(2) = 1;
%e A264596 S_3 = [0, 01, 1, 11], a(3) = 3;
%e A264596 S_4 = [0, 001, 01, 1, 11], a(4) = 1;
%e A264596 S_5 = [0, 001, 01, 1, 101, 11], a(5) = 4;
%e A264596 S_6 = [0, 001, 01, 011, 1, 101, 11], a(6) = 3;
%e A264596 S_7 = [0, 001, 01, 011, 1, 101, 11, 111], a(7) = 7;
%e A264596 S_8 = [0, 0001, 001, 01, 011, 1, 101, 11, 111], a(8) = 1;
%e A264596 ...
%p A264596 a:= proc(n) option remember; `if`(n=0, 0,
%p A264596       `if`(irem(n, 2, 'r')=0, a(r), a(r)+r+1))
%p A264596     end:
%p A264596 seq(a(n), n=0..100);  # _Alois P. Heinz_, Nov 19 2015
%t A264596 A264596[0]=0;A264596[n_]:=A264596[n]=A264596[Floor[n/2]]+Boole[OddQ[n]](Floor[n/2]+1);Array[A264596,100,0] (* _Paolo Xausa_, Nov 04 2023, after _Alois P. Heinz_ *)
%o A264596 (Python)
%o A264596 def A264596(n):
%o A264596     return sorted(format(i,'b')[::-1] for i in range(n+1)).index(format(n,'b')[::-1]) # _Chai Wah Wu_, Nov 22 2015
%Y A264596 Suggested by John Bodeen's A263856.
%Y A264596 Cf. A188215.
%K A264596 nonn,base
%O A264596 0,4
%A A264596 _N. J. A. Sloane_, Nov 19 2015
%E A264596 More terms from _Alois P. Heinz_, Nov 19 2015