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 A296000 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences. 23
 1, 3, 10, 37, 135, 493, 1800, 6572, 23996, 87614, 319895, 1167997, 4264577, 15570774, 56851829, 207576737, 757901769, 2767242128, 10103722287, 36890593353, 134694505577, 491795012865, 1795636233585, 6556206140806, 23937943641806, 87401941533192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  a(n)/a(n-1) -> 3.651188... (as in A295999).  Guide for the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0): A296000: a(0) = 1, a(1) = 3, b(0) = 2, limiting ratio of a(n)/a(n-1): A295999 A296001: a(0) = 1, a(1) = 2, b(0) = 3, limiting ratio of a(n)/a(n-1): A296002 A296003: a(0) = 2, a(1) = 4, b(0) = 1, limiting ratio of a(n)/a(n-1): A296004 A296005: a(0) = 2, a(1) = 3, b(0) = 1, limiting ratio of a(n)/a(n-1): A296006 LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. FORMULA Conjectures from Colin Barker, Dec 08 2017: (Start) G.f.: (1 - x)^2*(1 + x) / (1 - 4*x + x^2 + x^3 - x^8 + x^9). a(n) = 4*a(n-1) - a(n-2) - a(n-3) + a(n-8) - a(n-9) for n > 8. (End) EXAMPLE a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that a(2) = a(0)*b(1) + a(1)*b(0) = 10 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...) MATHEMATICA \$RecursionLimit = Infinity; mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &]; a[0] = 1; a[1] = 3; b[0] = 2; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, 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, 100}]  (* A296000 *) t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200] Take[RealDigits[Last[t], 10][[1]], 100]  (* A295999 *) CROSSREFS Cf. A295999, A296001. Sequence in context: A289615 A138807 A149043 * A242725 A151315 A164048 Adjacent sequences:  A295997 A295998 A295999 * A296001 A296002 A296003 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 04 2017 STATUS approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)