A184117 Lower s-Wythoff sequence, where s(n) = 2n + 1.

1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 95, 97, 98, 100, 101, 102, 104, 105, 107, 108, 110, 111, 112, 114, 115, 117, 118, 119, 121, 122, 124, 125, 126, 128, 129, 131, 132, 134, 135, 136, 138, 139, 141

Offset: 1

Author

Clark Kimberling, Jan 09 2011

Keywords

• nonn

Comments

Suppose that s(n) is a nondecreasing sequence of positive integers. The lower and upper s(n)-Wythoff sequences, a and b, are introduced here. Define

a(1) = 1; b(1) = s(1) + a(1); and for n>=2,

a(n) = least positive integer not in {a(1),...,a(n-1),b(1),...,b(n-1)},

b(n) = s(n) + a(n).

Clearly, a and b are complementary. If s(n)=n, then

a=A000201, the lower Wythoff sequence, and

b=A001950, the upper Wythoff sequence.

A184117 is chosen to represent the class of s-Wythoff sequences for which s is an arithmetic sequence given by s(n) = kn - r. Such sequences (lower and upper) are indexed in the OEIS as shown here:

n+1....A026273...A026274

n......A000201...A001950 (the classical Wythoff sequences)

2n+1...A184117...A184118

2n.....A001951...A001952

2n-1...A136119...A184119

3n+1...A184478...A184479

3n.....A184480...A001956

3n-1...A184482...A184483

3n-2...A184484...A184485

4n+1...A184486...A184487

4n.....A001961...A001962

4n-1...A184514...A184515

The pattern continues for A184516 to A184531.

s-Wythoff sequences for choices of s other than arithmetic sequences include these:

A184419 and A184420 (s = lower Wythoff sequence)

A184421 and A184422 (s = upper Wythoff sequence)

A184425 and A184426 (s = triangular numbers)

A184427 and A184428 (s = squares)

A036554 and A003159 (invariant and limiting sequences).

Examples

			s=(3,5,7,9,11,13,...);
a=(1,2,3,5,6,8,...);
b=(4,7,10,14,17,21,...).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Robbert Fokkink, Gerard Francis Ortega, and Dan Rust, Corner the Empress, arXiv:2204.11805 [math.CO], 2022. Mentions this sequence.

Crossrefs

Cf. A000201, A001950, A001951, A001952, A003159, A036554.

Programs

Mathematica

k=2; r=-1;
mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=k*n-r; a[1]=1; b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,1,n-1}]]];
Table[s[n],{n,30}]  (* s = A005408 except for initial 1 *)
Table[a[n],{n,100}] (* a = A184117 *)
Table[b[n],{n,100}] (* b = A184118 *)

PARI

A184117_upto(N,s(n)=2*n+1,a=[1],U=a)={while(a[#a]1&&U[2]==U[1]+1,U=U[^1]);a=concat(a,U[1]+1));a} \\ M. F. Hasler, Jan 07 2019

Formula

a(n) = A184118(n) - s(n). - M. F. Hasler, Jan 07 2019

Extensions

Removed an incorrect g.f., Alois P. Heinz, Dec 14 2012