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 A293765 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 41
 1, 3, 10, 20, 38, 67, 115, 193, 321, 528, 864, 1408, 2289, 3715, 6023, 9758, 15802, 25583, 41409, 67017, 108452, 175496, 283976, 459501, 743507, 1203039, 1946578, 3149650, 5096262, 8245947, 13342245, 21588229, 34930512, 56518780, 91449333, 147968155 (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) + 2 = 10; Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 2; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A293765 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622 (golden ratio), A293765. Sequence in context: A299692 A246520 A272764 * A295953 A005997 A213850 Adjacent sequences:  A293762 A293763 A293764 * A293766 A293767 A293768 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 29 2017 STATUS approved

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Last modified October 17 22:16 EDT 2019. Contains 328134 sequences. (Running on oeis4.)