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 A294367 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2
 1, 3, 9, 19, 37, 67, 117, 200, 335, 555, 912, 1491, 2429, 3948, 6407, 10387, 16829, 27253, 44121, 71415, 115579, 187039, 302665, 489753, 792469, 1282275, 2074799, 3357131, 5431989, 8789181, 14221233, 23010479, 37231779, 60242328, 97474179, 157716581 (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 = 12; Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 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] + 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}]  (* A294367 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622 (golden ratio), A293765. Sequence in context: A147158 A014540 A293058 * A146694 A146050 A147500 Adjacent sequences:  A294364 A294365 A294366 * A294368 A294369 A294370 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 29 2017 STATUS approved

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Last modified September 16 12:41 EDT 2019. Contains 327113 sequences. (Running on oeis4.)