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%I #36 Oct 24 2023 19:08:04
%S 1,2,4,5,7,8,9,10,12,13,15,16,17,18,20,21,23,24,26,27,29,30,31,32,34,
%T 35,37,38,39,40,42,43,45,46,48,49,51,52,53,54,56,57,59,60,61,62,64,65,
%U 67,68,69,70,72,73,75,76,77,78,80,81,83,84,86
%N a(n) = least k such that s(k) = n, where s = A026366.
%C Complement of A026368; also the rank transform (as at A187224) of A004526 (after removal of the initial two zeros). - _Clark Kimberling_, Mar 10 2011
%C Gives the positions of the 1's in A285431. - _Jeffrey Shallit_, Oct 21 2023
%C Conjecture: -1 < n*r - a(n) < 2 for n>=1, where r = (1 + sqrt(3))/2. - _Clark Kimberling_, Apr 29 2017
%H Clark Kimberling, <a href="/A026367/b026367.txt">Table of n, a(n) for n = 1..10000</a>
%H Robbert Fokkink, Gerard Francis Ortega, and Dan Rust, <a href="https://arxiv.org/abs/2204.11805">Corner the Empress</a>, arXiv:2204.11805 [math.CO], 2022. See Table 4.
%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2310.14252">Proof of Irvine's conjecture via mechanized guessing</a>, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
%F a(n) = a(n-1)+2 if n is in the set 2A+1, a(n) = a(n-1)+1 if n is not in 2A+1 where 2A+1 = {2a(1)+1,2a(2)+1,2a(3)+1,...} = {3,5,9,11,15,17,19,21,25,27,31,33,...}. a(n) = (1+sqrt(3))/2*n+O(1) For example, a(3) = a(2)+2 = 2+2 = 4 since 3 is in 2A+1. a(10) = a(9)+1 = 12+1 = 13 since 10 is not in 2A+1. - _Benoit Cloitre_, Apr 23 2008
%t seqA = Table[Ceiling[n/2], {n, 1, 180}] (* A004526 *)
%t seqB = Table[n, {n, 1, 80}]; (* A000027 *)
%t jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
%t Flatten@Position[#1, {_, 2}]} &[Sort@Flatten[{{#1, 1} & /@ seqA,
%t {#1, 2} & /@ seqB}, 1]];
%t limseqU = FixedPoint[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}]][[1]] (* A026367 *)
%t Complement[Range[Length[seqA]], limseqU] (* A026368 *)
%t (* _Peter J. C. Moses_, Mar 10 2011 *)
%t s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 0}}] &, {0}, 13] (* A285431 *)
%t Flatten[Position[s, 0]] (* A026368 *)
%t Flatten[Position[s, 1]] (* A026367 *)
%t (* _Clark Kimberling_, Apr 28 2017 *)
%Y Cf. A026368, A187224, A004526, A285431.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Complement sequence in first comment corrected by _Nathan Fox_, Mar 21 2014
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