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(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:
s-Wythoff sequences for choices of s other than arithmetic sequences include these:
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*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 *)
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 09 2011
EXTENSIONS
Removed an incorrect g.f., Alois P. Heinz, Dec 14 2012
STATUS
approved