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 A304502 Solution (c(n)) of the system of complementary equations defined in Comments. 3
 5, 11, 20, 26, 34, 41, 47, 53, 61, 68, 74, 83, 89, 95, 103, 110, 116, 124, 131, 137, 146, 152, 160, 167, 173, 179, 188, 194, 200, 209, 215, 223, 230, 236, 242, 250, 257, 263, 272, 278, 286, 293, 299, 305, 314, 320, 326, 335, 341, 349, 356, 362, 368, 377, 383 (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: a(n) = least new, b(n) = least new, c(n) = a(n) + 2*b(n), where "least new k" means the least positive integer not yet placed.  The three sequences partition the positive integers. Empirically, for all n >= 0:    1 <= 3*a(n) - 7*n <= 4,    5 <= 3*b(n) - 7*n <= 8,    4 <=   c(n) - 7*n <= 6. LINKS EXAMPLE a(0) = 1, b(0) = 2; c(0) = 1 + 2*2 = 5, so that a(1) = 3, so that b(1) = 4, so that c(1) = 11. MATHEMATICA z = 300; mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); a = {}; b = {}; c = {}; Do[AppendTo[a,    mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];   AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];   AppendTo[c, Last[a] + 2*Last[b]], {z}]; Take[a, 100] (* A304500 *) Take[b, 100] (* A304501 *) Take[c, 100] (* A304502 *) Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],   Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]}, Alignment -> ".",  Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}] (* Peter J. C. Moses, Apr 26 2018 *) CROSSREFS Cf. A304497, A304500, A304501. Sequence in context: A033913 A088124 A029456 * A190743 A331133 A110208 Adjacent sequences:  A304499 A304500 A304501 * A304503 A304504 A304505 KEYWORD nonn,easy AUTHOR Clark Kimberling, May 19 2018 STATUS approved

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Last modified May 30 15:21 EDT 2020. Contains 334726 sequences. (Running on oeis4.)