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 A294556 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
 1, 2, 13, 28, 57, 104, 183, 312, 523, 866, 1423, 2327, 3792, 6164, 10004, 16219, 26277, 42553, 68890, 111506, 180462, 292037, 472571, 764683, 1237332, 2002097, 3239515, 5241701, 8481308, 13723104, 22204510, 35927715, 58132329, 94060151, 152192590, 246252854 (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) + 3 = 13 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + 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}] (* A294556 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A294555 A031090 A358296 * A294559 A041017 A033837 Adjacent sequences: A294553 A294554 A294555 * A294557 A294558 A294559 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 15 2017 EXTENSIONS Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018 Definition corrected by Georg Fischer, Sep 27 2020 STATUS approved

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Last modified December 8 14:51 EST 2022. Contains 358695 sequences. (Running on oeis4.)