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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A288240 Decimal expansion of the limiting ratio of consecutive terms of A288230.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Examples

			1.9462903835027450512097395658148586704242126926733...

Crossrefs

Cf. A288230, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = Sqrt[5/2];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288230 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A288240 *)

A288241 Decimal expansion of the limiting ratio of consecutive terms of A288231.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Examples

			1.9479484718847796710315570516902584529350253894155...

Crossrefs

Cf. A288231, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = 4^(1/3);
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288231 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A288241 *)

A288242 Decimal expansion of the limiting ratio of consecutive terms of A288232.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Examples

			1.76404569251631086587666713867426335413247690520129811...

Crossrefs

Cf. A288232, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = 3E/5;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288232 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A288242 *)

A288935 Decimal expansion of the limiting ratio of consecutive terms of A288233.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Examples

			1.7639364499663486075051370496423665036228411...

Crossrefs

Cf. A288233, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = Sqrt[8/3];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288233 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A288935 *)

A289003 Decimal expansion of the limiting ratio of consecutive terms of A288234.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Examples

			1.755202484614384613489881050775533413444366653241137458292812482444...

Crossrefs

Cf. A288234, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = -1 + Sqrt[7];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288234 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289003 *)

A289033 Decimal expansion of the limiting ratio of consecutive terms of A288237.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Examples

			1.7548776666...

Crossrefs

Cf. A288237, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = -4/5 + Sqrt[11/4];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288237 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289033 *)

A289261 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=13/8.

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 91, 160, 281, 494, 871, 1537, 2711, 4782, 8437, 14885, 26258, 46320, 81712, 144145, 254277, 448555, 791273, 1395843, 2462330, 4343664, 7662429, 13516885, 23844416, 42062667, 74200520, 130893196, 230901729, 407321472, 718534172
Offset: 0

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Links

Crossrefs

Cf. A078140 (includes guide to related sequences), A289266.

Programs

  • Mathematica
    r = 13/8;
u = 1000; (* # initial terms from given series *)
v = 100;   (* # coefficients in reciprocal series *)
CoefficientList[Series[1/Sum[Floor[r*(k + 1)] (-x)^k, {k, 0, u}], {x, 0, v}], x]
LinearRecurrence[{2,-1,2,-2,1,-2,2},{1,3,5,9,17,30,52,91,160,281},40] (* Harvey P. Dale, Aug 04 2024 *)
  • PARI
    Vec((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x^4) / (1 - 2*x + x^2 - 2*x^3 + 2*x^4 - x^5 + 2*x^6 - 2*x^7) + O(x^50)) \\ Colin Barker, Jul 20 2017

Formula

G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = 13/8 and [ ] = floor.
G.f.: (1 - x)*(1 + x)^2*(1 + x^2)*(1 + x^4) / (1 - 2*x + x^2 - 2*x^3 + 2*x^4 - x^5 + 2*x^6 - 2*x^7). - Colin Barker, Jul 14 2017

A289262 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=11/7.

Original entry on oeis.org

1, 3, 5, 9, 18, 36, 71, 138, 268, 522, 1017, 1980, 3853, 7498, 14594, 28406, 55287, 107604, 209429, 407614, 793344, 1544090, 3005269, 5849172, 11384281, 22157298, 43124882, 83934214, 163361667, 317951804, 618831521, 1204435526, 2344200136, 4562530890
Offset: 0

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Links

Crossrefs

Cf. A078140 (includes guide to related sequences), A289267.

Programs

  • Mathematica
    r = 11/7;
u = 1000; (* # initial terms from given series *)
v = 100;   (* # coefficients in reciprocal series *)
CoefficientList[Series[1/Sum[Floor[r*(k + 1)] (-x)^k, {k, 0, u}], {x, 0, v}], x]
LinearRecurrence[{2,-1,2,-1,2,-2},{1,3,5,9,18,36,71,138,268},40] (* Harvey P. Dale, Jun 11 2024 *)
  • PARI
    Vec((1 + x)^2*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6) / (1 - 2*x + x^2 - 2*x^3 + x^4 - 2*x^5 + 2*x^6) + O(x^50)) \\ Colin Barker, Jul 20 2017

Formula

G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = 11/7 and [ ] = floor.
G.f.: (1 + x)^2*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6) / (1 - 2*x + x^2 - 2*x^3 + x^4 - 2*x^5 + 2*x^6). - Colin Barker, Jul 14 2017

A289266 Decimal expansion of the limiting ratio of consecutive terms of A289261.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Examples

			1.7640468687532034038928500563010585645579163733...

Crossrefs

Cf. A078140 (includes guide to related constants), A289261.

Programs

  • Mathematica
    z = 2000; r = 13/8;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289261 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289266 *)

Formula

Real root of the equation -2 + 2*x - x^2 + 2*x^3 - 2*x^4 + x^5 - 2*x^6 + x^7 = 0. - Vaclav Kotesovec, Aug 28 2021

A289267 Decimal expansion of the limiting ratio of consecutive terms of A289262.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Examples

			1.94630604272835654260927110687795006060927487947599...

Crossrefs

Cf. A078140 (includes guide to related constants), A289262.

Programs

  • Mathematica
    z = 2000; r = 11/7;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289262 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289267 *)

Formula

Real root of the equation 2 - 2*x + x^2 - 2*x^3 + x^4 - 2*x^5 + x^6 = 0. - Vaclav Kotesovec, Aug 28 2021
Previous Showing 21-30 of 39 results. Next

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