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%I #35 Apr 27 2022 11:12:38
%S 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,
%T 36,37,39,40,42,43,44,46,47,49,50,52,53,54,56,57,59,60,61,63,64,66,67,
%U 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
%N Lower s-Wythoff sequence, where s(n) = 2n + 1.
%C 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
%C a(1) = 1; b(1) = s(1) + a(1); and for n>=2,
%C a(n) = least positive integer not in {a(1),...,a(n-1),b(1),...,b(n-1)},
%C b(n) = s(n) + a(n).
%C Clearly, a and b are complementary. If s(n)=n, then
%C a=A000201, the lower Wythoff sequence, and
%C b=A001950, the upper Wythoff sequence.
%C 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:
%C n+1....A026273...A026274
%C n......A000201...A001950 (the classical Wythoff sequences)
%C 2n+1...A184117...A184118
%C 2n.....A001951...A001952
%C 2n-1...A136119...A184119
%C 3n+1...A184478...A184479
%C 3n.....A184480...A001956
%C 3n-1...A184482...A184483
%C 3n-2...A184484...A184485
%C 4n+1...A184486...A184487
%C 4n.....A001961...A001962
%C 4n-1...A184514...A184515
%C The pattern continues for A184516 to A184531.
%C s-Wythoff sequences for choices of s other than arithmetic sequences include these:
%C A184419 and A184420 (s = lower Wythoff sequence)
%C A184421 and A184422 (s = upper Wythoff sequence)
%C A184425 and A184426 (s = triangular numbers)
%C A184427 and A184428 (s = squares)
%C A036554 and A003159 (invariant and limiting sequences).
%H Alois P. Heinz, <a href="/A184117/b184117.txt">Table of n, a(n) for n = 1..1000</a>
%H Robbert Fokkink, Gerard Francis Ortega, and Dan Rust, <a href="https://arxiv.org/abs/2204.11805">Corner the Empress</a>, arXiv:2204.11805 [math.CO], 2022. Mentions this sequence.
%F a(n) = A184118(n) - s(n). - _M. F. Hasler_, Jan 07 2019
%e s=(3,5,7,9,11,13,...);
%e a=(1,2,3,5,6,8,...);
%e b=(4,7,10,14,17,21,...).
%t k=2; r=-1;
%t mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
%t s[n_]:=k*n-r; a[1]=1; b[n_]:=b[n]=s[n]+a[n];
%t a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,1,n-1}]]];
%t Table[s[n],{n,30}] (* s = A005408 except for initial 1 *)
%t Table[a[n],{n,100}] (* a = A184117 *)
%t Table[b[n],{n,100}] (* b = A184118 *)
%o (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
%Y Cf. A000201, A001950, A001951, A001952, A003159, A036554.
%K nonn
%O 1,2
%A _Clark Kimberling_, Jan 09 2011
%E Removed an incorrect g.f., _Alois P. Heinz_, Dec 14 2012
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