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 A294563 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 2, 6, 12, 24, 42, 73, 123, 205, 339, 555, 906, 1474, 2394, 3883, 6293, 10193, 16504, 26716, 43240, 69978, 113240, 183241, 296505, 479771, 776302, 1256100, 2032430, 3288559, 5321019, 8609609, 13930660, 22540302, 36470996, 59011333, 95482365 (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) - b(0) + 2 = 6 Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 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] - b[n - 2] + 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}]  (* A294563 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A222785 A053635 A054061 * A307211 A294545 A305104 Adjacent sequences:  A294560 A294561 A294562 * A294564 A294565 A294566 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 15 2017 STATUS approved

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Last modified August 19 07:13 EDT 2022. Contains 356216 sequences. (Running on oeis4.)