|
%I #87 Apr 26 2024 06:08:19
%S 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 18,28,20,24,16,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49,33,62,46,
%U 54,38,58,42,50,34,60,44,52,36,56,40,48,32,127,95,111,79,119,87,103,71
%N 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 A self-inverse permutation. Also a(n) = A054429(A059893(n)) = A059893(A054429(n)).
%C 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 Fixed points union { 0 } are in A290254. - _Alois P. Heinz_, Aug 24 2017
%H 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F 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 A000120(a(n)) = A000120(A054429(n)) = A023416(n) + 1 (conjectured). - _Ralf Stephan_, Oct 05 2003
%e a(9) = a(1001) = 1011 = 11.
%p a:= proc(n) local i, m, r; m, r:= n, 0;
%p for i from 0 while m>1 do r:= 2*r +1 -irem(m,2,'m') od;
%p r +2^i
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Feb 28 2015
%t 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 (PARI)
%o a(n)=local(v, l); v=binary(n); l=length(v); 2^(l-1)+sum(k=0, l-2, if(!v[k+2], 2^k))
%o log2(n) = log(n)/log(2)
%o b(n)=if(n<2, 1, if(n%2==0, b(n/2)+2^(floor(log2(n/2))+1), b((n-1)/2)+2^floor(log2((n-1)/2))))
%o (PARI)
%o { for (n=1, 1024, a=1; b=n; while (b>1, a=2*a + 1 - 2*frac(b/2); b=floor(b/2); ); write("b059894.txt", n, " ", a); ) } \\ _Harry J. Smith_, Jun 30 2009
%o (R)
%o maxrow <- 8 #by choice
%o a <- 1
%o for(m in 0:maxrow) for(k in 0:(2^m-1)){
%o a[2^(m+1) + 2*k ] <- a[2^m + k] + 2^(m+1)
%o a[2^(m+1) + 2*k + 1] <- a[2^m + k] + 2^m
%o }
%o a
%o # _Yosu Yurramendi_, Apr 05 2017
%o (Python)
%o def a(n): return int('1' + ''.join('0' if i=='1' else '1' for i in bin(n)[3:])[::-1], 2)
%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Aug 24 2017
%o (Python)
%o 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 {A000027, A054429, A059893, A059894} form a 4-group.
%Y Cf. A000120, A023416, A290253, A290254.
%K base,easy,nonn,look
%O 1,2
%A _Marc LeBrun_, Feb 06 2001
|
