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 A293406 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, 9, 18, 34, 60, 103, 174, 289, 476, 779, 1270, 2065, 3352, 5435, 8807, 14263, 23092, 37378, 60494, 97897, 158417, 256341, 414786, 671156, 1085972, 1757159, 2843163, 4600355, 7443552, 12043943, 19487532, 31531513, 51019084, 82550637, 133569762 (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 = 8; Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 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}]  (* A293406 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622 (golden ratio), A293076. Sequence in context: A093446 A256524 A210970 * A295862 A246695 A132920 Adjacent sequences:  A293403 A293404 A293405 * A293407 A293408 A293409 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 29 2017 STATUS approved

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Last modified June 1 18:43 EDT 2020. Contains 334762 sequences. (Running on oeis4.)