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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

Table of n, a(n) for n=0..25.

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.)