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 A298868 Solution (a(n)) of the system of 3 complementary equations in Comments. 8
 1, 4, 6, 8, 11, 14, 15, 17, 19, 21, 24, 26, 27, 29, 32, 33, 34, 37, 41, 42, 45, 46, 48, 52, 53, 54, 57, 58, 59, 61, 64, 67, 70, 72, 73, 74, 77, 79, 82, 83, 87, 90, 92, 93, 94, 96, 98, 100, 101, 104, 105, 107, 111, 113, 115, 118, 119, 120, 122, 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 k >= a(n) + n; 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. LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 EXAMPLE n: 0 1 2 3 4 5 6 7 8 9 a: 1 4 6 8 11 14 15 17 19 21 b: 2 5 7 10 12 16 20 22 25 28 c: 3 9 13 18 23 30 35 39 44 49 MATHEMATICA z = 400; mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0; Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]], AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]], AppendTo[c, Last[a] + Last[b]]}, {z}]; Take[a, 100] (* A298868 *) Take[b, 100] (* A298869 *) Take[c, 100] (* A298870 *) (* Peter J. C. Moses, Apr 08 2018 *) CROSSREFS Cf. A299634, A298869, A298870. Sequence in context: A003662 A132635 A182131 * A134779 A226250 A075325 Adjacent sequences: A298865 A298866 A298867 * A298869 A298870 A298871 KEYWORD nonn,easy AUTHOR Clark Kimberling, Apr 17 2018 STATUS approved

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Last modified August 10 15:57 EDT 2024. Contains 375057 sequences. (Running on oeis4.)