cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A296217 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) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 6, 26, 112, 484, 2088, 9008, 38862, 167658, 723308, 3120486, 13462360, 58079138, 250564260, 1080981064, 4663554414, 20119445656, 86799050160, 374467330636, 1615522076050, 6969664279584, 30068434774274, 129720849313094, 559639996988064, 2414391579204576
Offset: 0

Views

Author

Clark Kimberling, Dec 08 2017

Keywords

Comments

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.

Examples

			a(0) =1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 4
a(2) = a(1)*b(0) = 2
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

Links

Crossrefs

Cf. A296000, A296218.

Programs

  • Mathematica
    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 *)

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 08 2017

Keywords

Examples

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

Crossrefs

Cf. A296000, A296215, A296217.

Programs

  • Mathematica
    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 *)

A296216 Decimal expansion of the limiting ratio of terms in A296215.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 08 2017

Keywords

Examples

			A296215(n)/A296215(n-1) -> 3.667759717...
See A296000 for a guide to related sequences and limiting ratios.

Crossrefs

Cf. A296000, A296215, A296218.

Programs

  • Mathematica
    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 *)

A296221 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + 1, 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, 11, 40, 146, 533, 1946, 7105, 25941, 94714, 345812, 1262601, 4609907, 16831321, 61453163, 224372837, 819212023, 2991040928, 10920647625, 39872588647, 145579582824, 531528442330, 1940673819263, 7085631873740, 25870488153041, 94456241758347
Offset: 0

Views

Author

Clark Kimberling, Dec 09 2017

Keywords

Comments

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.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) + 1 = 11
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, ...)

Links

Crossrefs

Cf. A296000, A296222.

Programs

  • Mathematica
    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}] + 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}];  (* A296221 *)
Table[b[n], {n, 0, 20}]
t = N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
d = RealDigits[Last[t], 10][[1]] (* A296222 *)

Extensions

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

A296222 Decimal expansion of the limiting ratio of terms in A296221.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 09 2017

Keywords

Comments

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

Examples

			A296221(n)/A296221(n-1) -> 3.651119422238034...

Crossrefs

Cf. A296000, A296221.

Programs

  • Mathematica
    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}] + 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}];  (* A296221 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296222 *)

A296223 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - 1, 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, 9, 34, 124, 453, 1654, 6040, 22055, 80532, 294058, 1073735, 3920679, 14316124, 52274468, 190877084, 696976221, 2544966858, 9292793804, 33932079081, 123900951107, 452416889887, 1651973131976, 6032080786047, 22025781112962, 80425818360771
Offset: 0

Views

Author

Clark Kimberling, Dec 10 2017

Keywords

Comments

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.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) - 1 = 9
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)

Links

Crossrefs

Cf. A296000, A296224.

Programs

  • Mathematica
    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}] - 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}]  (* A296223 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296224 *)

A296224 Decimal expansion of the limiting ratio of terms in A296223.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 10 2017

Keywords

Comments

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

Examples

			3.651440004884...

Crossrefs

Cf. A296000, A296223.

Programs

  • Mathematica
    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}] - 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}]  (* A296223 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296224 *)

Formula

Equals lim_{n->oo} A296223(n)/A296223(n-1).

A296225 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + n, 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, 12, 44, 161, 588, 2147, 7839, 28621, 104498, 381533, 1393015, 5086038, 18569636, 67799608, 247543185, 903805055, 3299883119, 12048205018, 43989207775, 160609019998, 586399678681, 2141004179974, 7817021504815, 28540731390577, 104205079621096
Offset: 0

Views

Author

Clark Kimberling, Dec 10 2017

Keywords

Comments

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.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) + 2 = 12
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ...)

Links

Crossrefs

Cf. A296000, A296226.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
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}]  (* A296225 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296226 *)

A296226 Decimal expansion of the limiting ratio of terms in A296225.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 10 2017

Keywords

Comments

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

Examples

			3.651100534007...

Crossrefs

Cf. A296000, A296225.

Programs

  • Mathematica
    mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
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}]  (* A296225 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296226 *)

Formula

Equals lim_{n->oo} A296225(n)/A296225(n-1).

A296227 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

Original entry on oeis.org

1, 2, 8, 34, 146, 628, 2703, 11632, 50057, 215415, 927016, 3989317, 17167612, 73879038, 317930779, 1368182139, 5887829959, 25337665679, 109038016813, 469233798454, 2019298993572, 8689843823858, 37395841786394, 160929127296116, 692541811472532
Offset: 0

Views

Author

Clark Kimberling, Dec 10 2017

Keywords

Comments

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.

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) - 2 = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

Links

Crossrefs

Cf. A296000, A296228.

Programs

  • Mathematica
    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 *)
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