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

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, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 57, 59, 61, 62, 64, 65, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 90, 91, 93, 95, 96, 98, 99

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

This is the lower s-Wythoff sequence, where s(n)=n+1.

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:

s=(2,3,4,5,6,...);

a=(1,2,4,6,7,...)=A026273;

b=(3,5,8,11,13,...)=A026274.

Briefly: b=s+a, and a=mex="least missing".

From Michel Dekking, Mar 12 2018: (Start)

One has r*(n-2*r+3) = n*r-2r^2+3*r = (n+1)*r-2.

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.

(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).

This observation also gives an answer to Lenormand's question, and a simple proof of Mathar's conjecture in A059426.

(End)

Links

• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 3.

Crossrefs

Cf. A184117, A026274.

Programs

Mathematica

r=(1+Sqrt[5])/2;
a[n_]:=Floor[r*(n-2r+3)];
b[n_]:=Floor[r*r*(n+2r-3)];
Table[a[n],{n,200}]   (* A026273 *)
Table[b[n],{n,200}]   (* A026274 *)

Formula

a(n) = floor[r*(n-2*r+3)], where r=golden ratio.

b(n) = floor[(r^2)*(n+2*r-3)] = floor(n*A104457-A134972+1).

Extensions

Extended by Clark Kimberling, Jan 14 2011