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 A294546 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 5
 1, 2, 9, 19, 38, 69, 121, 207, 347, 575, 945, 1545, 2517, 4091, 6639, 10763, 17438, 28239, 45717, 73998, 119759, 193803, 313610, 507463, 821125, 1328642, 2149823, 3478523, 5628406, 9106991, 14735461, 23842518, 38578047, 62420635, 100998755, 163419465 (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, 8, 10, 11, 12, 13, 14, 15, 16, ...). 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] + n; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294546 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A031316 A335051 A173663 * A135207 A274853 A264670 Adjacent sequences:  A294543 A294544 A294545 * A294547 A294548 A294549 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 04 2017 STATUS approved

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Last modified May 17 07:36 EDT 2021. Contains 343966 sequences. (Running on oeis4.)