The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 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 = 1; a = 3; b = 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][], 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 EXTENSIONS Incorrect conjectured g.f. removed by Georg Fischer, Sep 23 2020 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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