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 A294541 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 5
 1, 2, 7, 14, 27, 49, 85, 144, 240, 396, 649, 1060, 1725, 2802, 4545, 7366, 11931, 19318, 31271, 50612, 81907, 132544, 214477, 347049, 561555, 908634, 1470220, 2378886, 3849139, 6228059, 10077233, 16305328, 26382598, 42687964, 69070601, 111758605 (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) = 7; Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, ...). 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]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294541 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A268347 A210728 A294533 * A294564 A068040 A200084 Adjacent sequences:  A294538 A294539 A294540 * A294542 A294543 A294544 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 04 2017 STATUS approved

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Last modified May 9 23:39 EDT 2021. Contains 343746 sequences. (Running on oeis4.)