A296005 -id:A296005 - cp's OEIS frontend

A296006 Decimal expansion of the limiting ratio of terms in A296005.

3, 1, 1, 4, 9, 8, 6, 4, 4, 7, 3, 9, 0, 3, 0, 2, 2, 4, 0, 2, 1, 1, 6, 2, 7, 0, 8, 7, 5, 5, 5, 4, 0, 3, 2, 8, 8, 3, 6, 0, 0, 9, 2, 9, 4, 5, 6, 0, 5, 9, 4, 2, 0, 4, 1, 3, 8, 8, 0, 8, 6, 4, 2, 5, 5, 2, 0, 9, 6, 6, 0, 7, 0, 5, 7, 3, 1, 7, 7, 4, 7, 9, 9, 6, 6, 2

Offset: 1

Clark Kimberling, Dec 07 2017

A296005(n)/A296005(n-1) -> 3.114986447390302...

See A296000 for a guide to related sequences and limiting ratios.

Cf. A296000, A296005.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 3; b[0] = 1;
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}];  (* A296005 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296006 *)

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.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 04 2017

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

			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, ...)

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A295999, A296001.

$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 *)

Incorrect conjectured g.f. removed by Georg Fischer, Sep 23 2020

cp's OEIS Frontend

A296006 Decimal expansion of the limiting ratio of terms in A296005.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

cp's OEIS Frontend

A296006 Decimal expansion of the limiting ratio of terms in A296005.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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.