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A341915 For any nonnegative number n with runs in binary expansion (r_1, ..., r_w), a(n) = Sum_{k = 1..w} 2^(r_1 + ... + r_k - 1).

%I A341915 #19 Jan 02 2022 07:25:45
%S A341915 0,1,3,2,5,7,6,4,9,13,15,11,10,14,12,8,17,25,29,21,23,31,27,19,18,26,
%T A341915 30,22,20,28,24,16,33,49,57,41,45,61,53,37,39,55,63,47,43,59,51,35,34,
%U A341915 50,58,42,46,62,54,38,36,52,60,44,40,56,48,32,65,97,113
%N A341915 For any nonnegative number n with runs in binary expansion (r_1, ..., r_w), a(n) = Sum_{k = 1..w} 2^(r_1 + ... + r_k - 1).
%C A341915 This sequence is a permutation of the nonnegative integers with inverse A341916.
%C A341915 This sequence has connections with A003188; here we compute partials sums of runs from left to right, there from right to left.
%H A341915 Rémy Sigrist, <a href="/A341915/b341915.txt">Table of n, a(n) for n = 0..8191</a>
%H A341915 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A341915 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A341915 a(n) = A059893(A003188(n)).
%F A341915 a(n) = Sum_{k = 1..A005811(n)} 2^((Sum_{m = 1..k} A101211(m))-1).
%F A341915 a(n) < 2^k for any n < 2^k.
%F A341915 A000120(a(n)) = A000120(A003188(n)) = A005811(n).
%e A341915 For n = 23,
%e A341915 - the binary representation of 23 is "10111",
%e A341915 - the corresponding run lengths are (1, 1, 3),
%e A341915 - so a(23) = 2^(1-1) + 2^(1+1-1) + 2^(1+1+3-1) = 19.
%t A341915 a[n_] := If[n == 0, 0, 2^((Length /@ Split[IntegerDigits[n, 2]] // Accumulate)-1) // Total];
%t A341915 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jan 02 2022 *)
%o A341915 (PARI) a(n) = { my (v=0); while (n, my (w=valuation(n+n%2,2)); n\=2^w; v=2^w*(1+v)); v/2 }
%Y A341915 Cf. A003188, A005811, A059893, A101211, A341916 (inverse), A341943 (fixed points).
%K A341915 nonn,look,base
%O A341915 0,3
%A A341915 _Rémy Sigrist_, Feb 23 2021