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 A297838 Solution (a(n)) of the system of 3 complementary equations in Comments. 5
 1, 4, 5, 7, 9, 12, 15, 16, 17, 20, 21, 25, 27, 28, 29, 33, 34, 35, 36, 39, 45, 46, 47, 48, 52, 56, 57, 58, 60, 61, 62, 64, 65, 67, 74, 75, 76, 78, 79, 80, 81, 87, 88, 94, 95, 97, 100, 102, 103, 104, 105, 106, 107, 108, 110, 114, 117, 123, 124, 125, 126, 127 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2: a(n) = least new; b(n) = least new > = a(n) + n + 1; c(n) = a(n) + b(n); where "least new k" means the least positive integer not yet placed. *** The sequences a,b,c partition the positive integers. *** Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67)) x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1. (The same limits occur in A298868 and A297469.) LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 EXAMPLE n:   0   1   2   3   4    5   6   7   8   9  10 a:   1   4   5   7   9   12  15  16  17  20  21 b:   2   6   8  11   14  19  22  24  26  30  32 c:   3  10  13  18   23  31  37  40  43  50  53 MATHEMATICA z=200; mex[list_, start_]:=(NestWhile[#+1&, start, MemberQ[list, #]&]); a={1}; b={2}; c={3}; n=0; Do[{n++;   AppendTo[a, mex[Flatten[{a, b, c}], If[Length[a]==0, 1, Last[a]]]],   AppendTo[b, mex[Flatten[{a, b, c}], Last[a]+n+1]],   AppendTo[c, Last[a]+Last[b]]}, {z}]; Take[a, 100] (* A297838 *) Take[b, 100] (* A298170 *) Take[c, 100] (* A298418 *) (* Peter J. C. Moses, Apr 23 2018 *) CROSSREFS Cf. A299634, A298868, A297469, A298170, A298418. Sequence in context: A286050 A047493 A285307 * A032360 A117150 A239276 Adjacent sequences:  A297835 A297836 A297837 * A297839 A297840 A297841 KEYWORD nonn,easy AUTHOR Clark Kimberling, Apr 25 2018 STATUS approved

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Last modified January 21 18:52 EST 2021. Contains 340352 sequences. (Running on oeis4.)