

A184117


Lower sWythoff sequence, where s(n) = 2n + 1.


54



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


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(n1),b(1),...,b(n1)},
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 sWythoff 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
2n1...A136119...A184119
3n+1...A184478...A184479
3n.....A184480...A001956
3n1...A184482...A184483
3n2...A184484...A184485
4n+1...A184486...A184487
4n.....A001961...A001962
4n1...A184514...A184515
The pattern continues for A184516 to A184531.
sWythoff 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).


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.


FORMULA

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


EXAMPLE

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


MATHEMATICA

k=2; r=1;
mex:=First[Complement[Range[1, Max[#1]+1], #1]]&;
s[n_]:=k*nr; a[1]=1; b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i], b[i]}, {i, 1, n1}]]];
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 *)


PROG

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


CROSSREFS

Cf. A000201, A001950, A001951, A001952, A003159, A036554.
Sequence in context: A026371 A187335 A161188 * A184624 A093001 A226721
Adjacent sequences: A184114 A184115 A184116 * A184118 A184119 A184120


KEYWORD

nonn


AUTHOR

Clark Kimberling, Jan 09 2011


EXTENSIONS

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


STATUS

approved



