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A026273 a(n) = least k such that s(k) = n, where s = A026272.

%I A026273 #25 Mar 28 2025 10:47:52
%S A026273 1,2,4,6,7,9,10,12,14,15,17,19,20,22,23,25,27,28,30,31,33,35,36,38,40,
%T A026273 41,43,44,46,48,49,51,53,54,56,57,59,61,62,64,65,67,69,70,72,74,75,77,
%U A026273 78,80,82,83,85,86,88,90,91,93,95,96,98,99
%N A026273 a(n) = least k such that s(k) = n, where s = A026272.
%C A026273 This is the lower s-Wythoff sequence, where s(n)=n+1.
%C A026273 See A184117 for the definition of lower and upper s-Wythoff sequences.  The first few terms of a and its complement, b=A026274, are obtained generated as follows:
%C A026273 s=(2,3,4,5,6,...);
%C A026273 a=(1,2,4,6,7,...)=A026273;
%C A026273 b=(3,5,8,11,13,...)=A026274.
%C A026273 Briefly:  b=s+a, and a=mex="least missing".
%C A026273 From _Michel Dekking_, Mar 12 2018: (Start)
%C A026273 One has r*(n-2*r+3) = n*r-2r^2+3*r = (n+1)*r-2.
%C A026273 So  a(n) = (n+1)*r-2, and we see that this sequence is simply the Beatty sequence of the golden ratio, shifted spatially and temporally. In other words: if w = A000201 = 1,3,4,6,8,9,11,12,14,...  is the lower Wythoff sequence, then a(n) = w(n+2) - 2.
%C A026273 (N.B. As so often, there is the 'offset 0 vs 1 argument', w = A000201 has offset 1; it would have been better to give (a(n)) offset 1, too).
%C A026273 This observation also gives an answer to Lenormand's question, and a simple proof of Mathar's conjecture in A059426.
%C A026273 (End)
%H A026273 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 3.
%F A026273 a(n) = floor[r*(n-2*r+3)], where r=golden ratio.
%F A026273 b(n) = floor[(r^2)*(n+2*r-3)] = floor(n*A104457-A134972+1).
%t A026273 r=(1+Sqrt[5])/2;
%t A026273 a[n_]:=Floor[r*(n-2r+3)];
%t A026273 b[n_]:=Floor[r*r*(n+2r-3)];
%t A026273 Table[a[n],{n,200}]   (* A026273 *)
%t A026273 Table[b[n],{n,200}]   (* A026274 *)
%Y A026273 Cf. A184117, A026274.
%K A026273 nonn
%O A026273 1,2
%A A026273 _Clark Kimberling_
%E A026273 Extended by _Clark Kimberling_, Jan 14 2011