A296217 -id:A296217 - cp's OEIS frontend

A296218 Decimal expansion of the limiting ratio of terms in A296217.

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

Offset: 1

Clark Kimberling, Dec 08 2017

			A296217(n)/A296217(n-1) -> 4.31418696340...
See A296000 for a guide to related sequences and limiting ratios.

Cf. A296000, A296215, A296217.

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296217 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]]    (* A296218 *)

A296215 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + 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, 6, 24, 87, 321, 1176, 4314, 15822, 58032, 212847, 780672, 2863317, 10501959, 38518662, 141277197, 518170812, 1900526031, 6970672818, 25566752964, 93772706622, 343935755925, 1261473710904, 4626782461218, 16969926331719, 62241612204120, 228287277978756

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, b(2) = 5
a(2) = a(1)*b(0) = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...)

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

Cf. A296000, A296217.

a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296215 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296216 *)

cp's OEIS Frontend

A296218 Decimal expansion of the limiting ratio of terms in A296217.

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A296215 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + 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.

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cp's OEIS Frontend

A296218 Decimal expansion of the limiting ratio of terms in A296217.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A296215 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + 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

A296215 Solution of the complementary equation a(n) = a(1)b(n-2) + a(2)b(n-3) + ... + 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.