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 A293358 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 14
 1, 3, 8, 16, 30, 53, 92, 155, 258, 425, 696, 1135, 1846, 2998, 4862, 7879, 12761, 20661, 33444, 54128, 87596, 141749, 229371, 371147, 600546, 971722, 1572299, 2544053, 4116385, 6660472, 10776892, 17437400, 28214329, 45651767, 73866135, 119517942 (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: A293358:  a(n) = a(n-1) + a(n-2) + b(n-1) A293406:  a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A293765:  a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A293766:  a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A293767:  a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294365:  a(n) = a(n-1) + a(n-2) + b(n-1) + n A294366:  a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294367:  a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294368:  a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. 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) = 8; Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...) 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]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A293358 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622 (golden ratio), A293076. Sequence in context: A009439 A000233 A002624 * A227265 A295960 A068039 Adjacent sequences:  A293355 A293356 A293357 * A293359 A293360 A293361 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 29 2017 STATUS approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)