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 A294543 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 2, 9, 18, 35, 62, 107, 181, 301, 496, 812, 1324, 2153, 3495, 5667, 9183, 14872, 24078, 38974, 63077, 102077, 165181, 267286, 432496, 699812, 1132339, 1832183, 2964555, 4796772, 7761362, 12558170, 20319570, 32877779, 53197389, 86075209, 139272640 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, so that b(1) = 4 (least "new number"); a(2) = a(1) + a(0) + b(1) + 2 = 9. Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...). MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; b = 2; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 2; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294543 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A103256 A028881 A294535 * A295956 A296843 A200085 Adjacent sequences:  A294540 A294541 A294542 * A294544 A294545 A294546 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 04 2017 STATUS approved

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Last modified April 13 18:46 EDT 2021. Contains 342939 sequences. (Running on oeis4.)