A056539 - OEIS

%I #23 Apr 29 2017 05:38:30

%S 0,1,2,3,6,5,4,7,14,9,10,13,12,11,8,15,30,17,22,25,26,21,18,29,28,19,

%T 20,27,24,23,16,31,62,33,46,49,54,41,38,57,58,37,42,53,50,45,34,61,60,

%U 35,44,51,52,43,36,59,56,39,40,55,48,47,32,63,126,65,94,97,110,81,78

%N Self-inverse permutation: reverse the bits in binary expansion of n and also complement them (0->1, 1->0) if the run count (A005811) is even.

%H Paul Tek, <a href="/A056539/b056539.txt">Table of n, a(n) for n = 0..8191</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(2n) = A036044(2n), a(2n+1) = A030101(2n+1). - _Antti Karttunen_, Feb 14 2003

%e n: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

%e binary expansion: 0, 1, 10, 11, 100, 101, 110, 111,1000,1001,1010,1011,1100,1101,1110,1111

%e reversed/complemented: 0, 1, 10, 11, 110, 101, 100, 111,1110,1001,1010,1101,1100,1011,1000,1111

%p [seq(runcounts2binexp(reverse(binexp2runcounts(j))),j=0..511)];

%p runcounts2binexp := proc(c) local i,e,n; n := 0; for i from 1 to nops(c) do e := c[i]; n := ((2^e)*n) + ((i mod 2)*((2^e)-1)); od; RETURN(n); end;

%p binexp2runcounts := proc(nn) local n,a,p,c; n := nn; a := []; p := (`mod`(n,2)); c := 0; while(n > 0) do c := c+1; n := floor(n/2); if((`mod`(n,2)) <> p) then a := [c,op(a)]; c := 0; p := (`mod`(p+1,2)); fi; od; RETURN(a); end;

%p # reverse given in A056538

%o (Python)

%o def a005811(n): return bin(n^(n>>1))[2:].count("1")

%o def a(n):

%o     if n==0: return 0

%o     x=bin(n)[2:][::-1]

%o     if a005811(n)%2==1: return int(x, 2)

%o     z=''.join('1' if i == '0' else '0' for i in x)

%o     return int(z, 2) # _Indranil Ghosh_, Apr 29 2017

%Y Cf. A054429.

%Y When restricted to A014486 induces another permutation, A057164. A105726 is a "deep" variant.

%K base,nonn,look

%O 0,3

%A _Antti Karttunen_, Jun 20, 2000