A296000 -id:A296000 - cp's OEIS frontend

A296228 Decimal expansion of the limiting ratio of terms in A296227.

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

Offset: 1

Clark Kimberling, Dec 10 2017

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

			4.3033963031...

Cf. A296000, A296227.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = - 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, 200}]  (* A296227 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296228 *)

Equals lim_{n->oo} A296227(n)/A296227(n-1).

A296219 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2), 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, 17, 21, 25, 29, 33, 38, 45, 49, 53, 57, 61, 66, 73, 77, 82, 89, 93, 98, 105, 109, 114, 121, 125, 130, 137, 141, 145, 150, 157, 161, 165, 169, 173, 178, 185, 189, 194, 201, 205, 210, 217, 221, 226, 233, 237, 242, 249, 253, 257, 262, 269, 273, 277

Offset: 0

Clark Kimberling, Dec 08 2017

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
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, 18, ...)

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

Cf. A296000.

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 500}];  (* A296219 *)
Table[b[n], {n, 0, 20}]

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

cp's OEIS Frontend

A296228 Decimal expansion of the limiting ratio of terms in A296227.

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A296219 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

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cp's OEIS Frontend

A296228 Decimal expansion of the limiting ratio of terms in A296227.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A296219 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), 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

A296219 Solution of the complementary equation a(n) = a(0)b(n-1) + a(1)b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.