A184516 -id:A184516 - cp's OEIS frontend

A184517 Upper s-Wythoff sequence, where s=4n-2. Complement of A184516.

3, 8, 14, 19, 24, 29, 35, 40, 45, 50, 55, 61, 66, 71, 76, 82, 87, 92, 97, 103, 108, 113, 118, 124, 129, 134, 139, 144, 150, 155, 160, 165, 171, 176, 181, 186, 192, 197, 202, 207, 213, 218, 223, 228, 234, 239, 244, 249, 254, 260, 265, 270, 275, 281, 286, 291, 296, 302, 307, 312, 317, 323, 328, 333, 338, 343, 349, 354, 359, 364, 370, 375, 380, 385, 391, 396, 401, 406, 412, 417, 422, 427, 432, 438, 443, 448, 453, 459, 464, 469, 474, 480, 485, 490, 495, 501, 506

Offset: 1

Clark Kimberling, Jan 16 2011

nonn

See A184117 for the definition of lower and upper s-Wythoff sequences.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Clark Kimberling, Beatty Sequences and Wythoff Sequences, Generalized, Fibonacci Quart. 49 (2011), no. 3, 195-200.

Cf. A184117, A184516.

[Floor((3+Sqrt(5))*(n - 1/(1+Sqrt(5)))): n in [1..100]]; // G. C. Greubel, Nov 16 2018

k = 4; r = 2; d = Sqrt[4 + k^2];
a[n_] := Floor[(1/2) (d + 2 - k) (n + r/(d + 2))];
b[n_] := Floor[(1/2) (d + 2 + k) (n - r/(d + 2))];
Table[a[n], {n, 120}] (* A184516 *)
Table[b[n], {n, 120}] (* A184517 *)
(* alternate program *)
Table[Ceiling[(2 n - 1) GoldenRatio^2], {n, 1, 120}] (* Jon Maiga, Nov 15 2018 *)

vector(100, n, floor((3+sqrt(5))*(n - 1/(1+sqrt(5))))) \\ G. C. Greubel, Nov 16 2018

[floor((3+sqrt(5))*(n - 1/(1+sqrt(5)))) for n in (1..100)] # G. C. Greubel, Nov 16 2018

a(n) = ceiling((2*n-1)*phi^2), where phi = A001622. - Jon Maiga, Nov 15 2018

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

Original entry on oeis.org

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

Clark Kimberling, Jan 09 2011

nonn

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).

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

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.

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

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 *)

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

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

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

cp's OEIS Frontend

A184517 Upper s-Wythoff sequence, where s=4n-2. Complement of A184516.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula