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 A293767 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2
 1, 3, 7, 14, 26, 47, 81, 137, 228, 376, 616, 1006, 1637, 2659, 4313, 6990, 11322, 18332, 29675, 48029, 77727, 125780, 203533, 329340, 532901, 862270, 1395201, 2257502, 3652735, 5910270, 9563039, 15473344, 25036419, 40509800, 65546257, 106056096 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4: Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.  See A293358 for a guide to related sequences. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that a(2) = a(1) + a(0) + b(1) - 1 = 7; Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; b = 2; b = 4; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - 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}]  (* A293767 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622 (golden ratio), A293765. Sequence in context: A014153 A001924 A079921 * A014168 A132109 A317779 Adjacent sequences:  A293764 A293765 A293766 * A293768 A293769 A293770 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 29 2017 STATUS approved

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Last modified November 22 13:47 EST 2019. Contains 329393 sequences. (Running on oeis4.)