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A329303 If the run lengths in binary expansion of n are (r(1), ..., r(w)), then the run lengths in binary expansion of a(n) are (r(1), r(3), r(5), ..., r(6), r(4), r(2)).

%I A329303 #22 Dec 02 2019 21:15:57
%S A329303 0,1,2,3,4,5,6,7,8,11,10,9,12,13,14,15,16,23,20,19,22,21,18,17,24,27,
%T A329303 26,25,28,29,30,31,32,47,40,39,44,43,36,35,46,41,42,45,38,37,34,33,48,
%U A329303 55,52,51,54,53,50,49,56,59,58,57,60,61,62,63,64,95,80,79
%N A329303 If the run lengths in binary expansion of n are (r(1), ..., r(w)), then the run lengths in binary expansion of a(n) are (r(1), r(3), r(5), ..., r(6), r(4), r(2)).
%C A329303 This sequence is a permutation of the nonnegative integers that preserves the binary length as well as the number of runs. See A330091 for the inverse.
%H A329303 Rémy Sigrist, <a href="/A329303/b329303.txt">Table of n, a(n) for n = 0..8192</a>
%H A329303 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A329303 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A329303 If n has w binary runs, then a^A003558(w-1)(n) = n (where a^k denotes the k-th iterate of the sequence).
%e A329303 For n = 19999:
%e A329303 - the binary representation of 19999 is "100111000011111",
%e A329303 - the corresponding run lengths are (1, 2, 3, 4, 5),
%e A329303 - hence the run lengths of a(n) are (1, 3, 5, 4, 2),
%e A329303 - and its binary representation is "100011111000011",
%e A329303 - so a(n) = 18371.
%o A329303 (PARI) torl(n) = { my (rr=[]); while (n, my (r=valuation(n+(n%2),2)); rr = concat(r, rr); n\=2^r); rr }
%o A329303 shuffle(v) = { my (w=vector(#v), o=0, e=#v+1); for (k=1, #v, w[if (k%2, o++, e--)]=v[k]); w }
%o A329303 fromrl(rr) = { my (v=0); for (k=1, #rr, v = (v+(k%2))*2^rr[k]-(k%2)); v }
%o A329303 a(n) = fromrl(shuffle(torl(n)))
%Y A329303 See A330081 for a similar sequence.
%Y A329303 Cf. A003558, A194959, A330091 (inverse).
%K A329303 nonn,base
%O A329303 0,3
%A A329303 _Rémy Sigrist_, Dec 01 2019