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 A184117 Lower s-Wythoff 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(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). 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]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

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Last modified December 9 17:12 EST 2022. Contains 358702 sequences. (Running on oeis4.)