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A059894 Complement and reverse the order of all but the most significant bit in binary expansion of n. n = 1ab..yz -> 1ZY..BA = a(n), where A = 1-a, B = 1-b, ... .

%I A059894 #97 Dec 08 2024 12:58:49
%S A059894 1,3,2,7,5,6,4,15,11,13,9,14,10,12,8,31,23,27,19,29,21,25,17,30,22,26,
%T A059894 18,28,20,24,16,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49,33,62,46,
%U A059894 54,38,58,42,50,34,60,44,52,36,56,40,48,32,127,95,111,79,119,87,103,71
%N A059894 Complement and reverse the order of all but the most significant bit in binary expansion of n. n = 1ab..yz -> 1ZY..BA = a(n), where A = 1-a, B = 1-b, ... .
%C A059894 A self-inverse permutation. Also a(n) = A054429(A059893(n)) = A059893(A054429(n)).
%C A059894 a(n) is the viabin number of the integer partition that is conjugate to the integer partition with viabin number n. Example: a(9) = 11. Indeed, 9 and 11 are the viabin numbers of the conjugate partitions [2,1,1] and [3,1], respectively. For the definition of viabin number see comment in A290253. - _Emeric Deutsch_, Aug 23 2017
%C A059894 Fixed points union { 0 } are in A290254. - _Alois P. Heinz_, Aug 24 2017
%H A059894 Alois P. Heinz, <a href="/A059894/b059894.txt">Table of n, a(n) for n = 1..8191</a> (first 1024 terms from Harry J. Smith)
%H A059894 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A059894 a(1) = 1, a(2n) = a(n) + 2^(floor(log_2(n))+1), a(2n+1) = a(n) + 2^floor(log_2(n)) (conjectured). - _Ralf Stephan_, Aug 21 2003
%F A059894 A000120(a(n)) = A000120(A054429(n)) = A023416(n) + 1 (conjectured). - _Ralf Stephan_, Oct 05 2003
%e A059894 a(9) = a(1001_2) = 1011_2 = 11.
%p A059894 a:= proc(n) local i, m, r; m, r:= n, 0;
%p A059894       for i from 0 while m>1 do r:= 2*r +1 -irem(m,2,'m') od;
%p A059894       r +2^i
%p A059894     end:
%p A059894 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 28 2015
%t A059894 Map[FromDigits[#, 2] &@ Flatten@ MapAt[Reverse, TakeDrop[IntegerDigits[#, 2], 1], -1] &, Flatten@ Table[Range[2^(n + 1) - 1, 2^n, -1], {n, 0, 6}]] (* _Michael De Vlieger_, Aug 23 2017 after _Harvey P. Dale_ at A054429 *)
%o A059894 (PARI) a(n)= my(b=binary(n)); 2^#b-1-fromdigits(Vecrev(b[2..#b]), 2); \\ _Ruud H.G. van Tol_, Nov 17 2024
%o A059894 (R)
%o A059894 maxrow <- 8 #by choice
%o A059894 a <- 1
%o A059894 for(m in 0:maxrow) for(k in 0:(2^m-1)){
%o A059894 a[2^(m+1) + 2*k    ] <- a[2^m + k] + 2^(m+1)
%o A059894 a[2^(m+1) + 2*k + 1] <- a[2^m + k] + 2^m
%o A059894 }
%o A059894 a
%o A059894 # _Yosu Yurramendi_, Apr 05 2017
%o A059894 (Python)
%o A059894 def a(n): return int('1' + ''.join('0' if i=='1' else '1' for i in bin(n)[3:])[::-1], 2)
%o A059894 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Aug 24 2017
%o A059894 (Python)
%o A059894 def A059894(n): return n if n <= 1 else -int((s:=bin(n)[-1:2:-1]),2)-1+2**(len(s)+1) # _Chai Wah Wu_, Feb 04 2022
%Y A059894 {A000027, A054429, A059893, A059894} form a 4-group.
%Y A059894 Cf. A000120, A023416, A290253, A290254.
%K A059894 base,easy,nonn,look
%O A059894 1,2
%A A059894 _Marc LeBrun_, Feb 06 2001