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A266089 Lexicographically smallest permutation of natural numbers such that in binary representation the number of ones of adjacent terms differ exactly by one.

%I A266089 #23 Mar 22 2025 11:45:50
%S A266089 0,1,3,2,5,4,6,7,9,8,10,11,12,13,15,14,17,16,18,19,20,21,23,22,24,25,
%T A266089 27,26,29,28,30,31,39,35,33,32,34,37,36,38,40,41,43,42,45,44,46,47,51,
%U A266089 49,48,50,53,52,54,55,57,56,58,59,60,61,63,62,71,67,65
%N A266089 Lexicographically smallest permutation of natural numbers such that in binary representation the number of ones of adjacent terms differ exactly by one.
%H A266089 Reinhard Zumkeller, <a href="/A266089/b266089.txt">Table of n, a(n) for n = 0..10000</a>
%H A266089 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A266089 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A266089 A266161(n) = A000120(a(n)).
%F A266089 abs(A000120(a(n+1)) - A000120(a(n))) = abs(A266161(n+1) - A266161(n)) = 1.
%e A266089 .   n |  a(n) | A007088(a(n)) | A266161(n)
%e A266089 . ----+-------+---------------+------------
%e A266089 .   0 |    0  |            0  |         0
%e A266089 .   1 |    1  |            1  |         1
%e A266089 .   2 |    3  |           11  |         2
%e A266089 .   3 |    2  |           10  |         1
%e A266089 .   4 |    5  |          101  |         2
%e A266089 .   5 |    4  |          100  |         1
%e A266089 .   6 |    6  |          110  |         2
%e A266089 .   7 |    7  |          111  |         3
%e A266089 .   8 |    9  |         1001  |         2
%e A266089 .   9 |    8  |         1000  |         1
%e A266089 .  10 |   10  |         1010  |         2
%e A266089 .  11 |   11  |         1011  |         3
%e A266089 .  12 |   12  |         1100  |         2
%e A266089 .  13 |   13  |         1101  |         3
%e A266089 .  14 |   15  |         1111  |         4
%e A266089 .  15 |   14  |         1110  |         3
%e A266089 .  16 |   17  |        10001  |         2  .
%t A266089 a[0] = 0; a[n_] := a[n] = Module[{bw = DigitCount[a[n - 1], 2, 1], k = 1}, While[!FreeQ[Array[a, n - 1], k] || Abs[DigitCount[k, 2, 1] - bw] != 1, k++]; k]; Array[a, 100, 0] (* _Amiram Eldar_, Jul 18 2023 *)
%o A266089 (Haskell)
%o A266089 import Data.List (delete)
%o A266089 a266089 n = a266089_list !! n
%o A266089 a266089_list = 0 : f 0 (zip [1..] $ tail a000120_list) where
%o A266089    f x zws = g zws where
%o A266089      g (yw@(y, w) : yws) | abs (x - w) /= 1 = g yws
%o A266089                          | otherwise = y : f w (delete yw zws)
%Y A266089 Cf. A000120, A007088, A108971, A266154 (inverse), A266161.
%K A266089 nonn,base
%O A266089 0,3
%A A266089 _Reinhard Zumkeller_, Dec 22 2015