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 A299636 Solution (c(n)) of the system of 3 complementary equations in Comments. 3
 3, 9, 16, 19, 22, 28, 36, 41, 48, 57, 61, 66, 74, 77, 83, 89, 94, 97, 101, 103, 108, 115, 121, 130, 133, 136, 139, 146, 154, 157, 161, 166, 171, 178, 183, 191, 200, 209, 214, 217, 222, 229, 238, 241, 244, 248, 253, 257, 265, 275, 282, 290, 295, 298, 306, 317 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2: a(n) = least new k >= 2*b(n-1); b(n) = least new k; 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 = 11/6. Conjectures: a(n) - 2*n*x = 0 for infinitely many n; b(n) - n*x = 0 for infinitely many n; c(n) - 3*n*x = 0 for infinitely many n; (a(n) - 2*n*x) is unbounded below and above; (b(n) - n*x) is unbounded below and above; (c(n) - 3*n*x) is unbounded below and above; *** Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures: 12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b); 2 occurs infinitely many times in d(c). 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 10 12 14 17 23 26 30 37 b: 2 5 6 7 8 11 13 15 18 20 c: 3 9 16 19 22 28 36 41 48 57 MATHEMATICA z = 1000; mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; Do[{ AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]], AppendTo[b, mex[Flatten[{a, b, c}], 1]], AppendTo[c, Last[a] + Last[b]]}, {z}]; Take[a, 100] (* A299634 *) Take[b, 100] (* A299635 *) Take[c, 100] (* A299636 *) (* Peter J. C. Moses, Apr 08 2018 *) CROSSREFS Cf. A299634, A299635. Sequence in context: A336970 A366051 A070066 * A271491 A359618 A214644 Adjacent sequences: A299633 A299634 A299635 * A299637 A299638 A299639 KEYWORD nonn,easy AUTHOR Clark Kimberling, Apr 17 2018 STATUS approved

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Last modified September 10 09:32 EDT 2024. Contains 375786 sequences. (Running on oeis4.)