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.

Previous Showing 31-39 of 39 results.

A289848 Decimal expansion of the limiting ratio of consecutive terms of A289245.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2017

Keywords

Examples

			2.5154769107727217467214881264679194776786944...

Crossrefs

Cf. A078140 (includes guide to related constants), A289245.

Programs

  • Mathematica
    z = 2000;
u = CoefficientList[Series[1/Sum[Floor[-1 + (1 + GoldenRatio)*(k + 1)] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289245*)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289848 *)

A289912 Coefficients of 1/(Sum_{k>=0} round((k+1)*r)(-x)^k), where r = sqrt(2).

Original entry on oeis.org

1, 3, 5, 9, 18, 35, 66, 124, 234, 441, 829, 1557, 2925, 5496, 10325, 19394, 36429, 68428, 128532, 241425, 453475, 851775, 1599910, 3005145, 5644626, 10602419, 19914742, 37406262, 70260933, 131972522, 247886635, 465610427, 874565375, 1642713630, 3085541851
Offset: 0

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Crossrefs

Cf. A078140 (includes guide to related sequences), A289913.

Programs

  • Mathematica
    z = 100; r = Sqrt[2];
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289912 *)
v = N[u[[z]]/u[[z - 1]], 200]
d = RealDigits[v, 10][[1]] (* A289913 *)

Formula

G.f.: 1/(Sum_{k>=0} round((k+1)*r)(-x)^k), where r = sqrt(2).

A289913 Decimal expansion of the limiting ratio of consecutive terms of A289912.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Examples

			1.87831998264762017350476950388676342028750759950872900...

Crossrefs

Cf. A078140 (includes guide to related constants), A289912.

Programs

  • Mathematica
    z = 100; r = Sqrt[2];
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289912 *)
v = N[u[[z]]/u[[z - 1]], 200]
d = RealDigits[v, 10][[1]] (* A289913 *)

A289914 Coefficients of 1/(Sum_{k>=0} round((k+1)*r)(-x)^k), where r = 7/5.

Original entry on oeis.org

1, 3, 5, 9, 18, 35, 66, 124, 234, 441, 830, 1563, 2944, 5544, 10440, 19661, 37026, 69727, 131310, 247284, 465686, 876981, 1651534, 3110175, 5857092, 11030096, 20771916, 39117745, 73666674, 138729339, 261255578, 491997420, 926531266, 1744846929, 3285901854
Offset: 0

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Links

Crossrefs

Cf. A078140 (includes guide to related sequences), A289915.

Programs

  • Mathematica
    z = 2000; r = 7/5;
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289914  *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289915 *)
  • PARI
    Vec((1+x)^2*(1-x+x^2-x^3+x^4) / (1-2*x+x^2-2*x^3+x^4) + O(x^50)) \\ Colin Barker, Jul 20 2017

Formula

G.f.: 1/(Sum_{k>=0} round((k+1)*r)(-x)^k), where r = 7/5.
From Colin Barker, Jul 19 2017: (Start)
G.f.: (1+x)^2*(1-x+x^2-x^3+x^4) / (1-2*x+x^2-2*x^3+x^4).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) for n>3.
(End)

A289915 Decimal expansion of the limiting ratio of consecutive terms of A289914.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Examples

			1.883203505913525864168947465362055090560951328672239179570777...

Links

Crossrefs

Cf. A078140 (includes guide to related constants), A289914.

Programs

  • Mathematica
    z = 2000; r = 7/5;
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289914  *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289915 *)
First[RealDigits[(1 + Sqrt[2] + Sqrt[2*Sqrt[2] - 1])/2, 10, 100]] (* Paolo Xausa, Feb 08 2024 *)

Formula

Equals (1 + sqrt(2) + sqrt(2*sqrt(2) - 1))/2. - Vaclav Kotesovec, Aug 27 2021
Largest real root of x^4 - 2*x^3 + x^2 - 2*x + 1. - Linas Vepstas, Feb 06 2024

A289916 Coefficients of 1/(Sum_{k>=0} round((k+1)*r)(-x)^k), where r = 9/7.

Original entry on oeis.org

1, 3, 5, 8, 13, 22, 39, 69, 120, 206, 353, 607, 1046, 1803, 3106, 5348, 9208, 15856, 27306, 47025, 80982, 139457, 240155, 413566, 712196, 1226463, 2112073, 3637166, 6263503, 10786276, 18574872, 31987488, 55085136, 94861220, 163358969, 281317834, 484452887
Offset: 0

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Links

Crossrefs

Cf. A078140 (includes guide to related sequences), A289917.

Programs

  • Mathematica
    z = 2000; r = 9/7;
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289916  *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289917 *)
  • PARI
    Vec((1+x)^2*(1-x+x^2-x^3+x^4-x^5+x^6) / ((1-x+x^2)*(1-x-x^2-x^3+x^4)) + O(x^50)) \\ Colin Barker, Jul 20 2017

Formula

G.f.: 1/(Sum_{k>=0} round((k+1)*r)(-x)^k), where r = 9/7.
From Colin Barker, Jul 19 2017: (Start)
G.f.: (1+x)^2*(1-x+x^2-x^3+x^4-x^5+x^6) / ((1-x+x^2)*(1-x-x^2-x^3+x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) + 2*a(n-5) - a(n-6) for n>5.
(End)

A288135 Coefficients of 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = sqrt(7/3) and [ ] = floor.

Original entry on oeis.org

1, 3, 5, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 147456, 294911, 589818, 1179628, 2359242, 4718457, 9436860, 18873612, 37747008, 75493584, 150986304, 301970880, 603938304, 1207869696, 2415725568, 4831423488, 9662791680
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Crossrefs

Cf. A078140 (includes guide to related sequences).

Programs

  • Mathematica
    r = Sqrt[7/3];
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]

Formula

G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = sqrt(7/3) and [ ] = floor.

A289245 Coefficients of 1/(Sum_{k>=0} [-1 + (k+1)*r](-x)^k), where r = (3 + sqrt(5))/2 = 1 + golden ratio and [ ] = floor.

Original entry on oeis.org

1, 4, 10, 25, 64, 162, 408, 1027, 2584, 6500, 16351, 41132, 103468, 260272, 654709, 1646907, 4142758, 10421013, 26213819, 65940258, 165871197, 417245167, 1049570586, 2640170577, 6641288127, 16706006942, 42023574736, 105709331958, 265909383794, 668888915293
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Crossrefs

Cf. A001622, A078140, A289848.

Programs

  • Mathematica
    CoefficientList[Series[1/Sum[Floor[-1 + (k + 1)*(1 + GoldenRatio)] (-x)^k, {k, 0, 100}], {x, 0, 50}], x]

Formula

G.f.: 1/(Sum_{k>=0} [-1 + (k+1)*r](-x)^k), where r = (3 + sqrt(5))/2 = 1 + golden ratio and [ ] = floor.

A289246 Coefficients in the expansion of 1/Sum_{k >= 0} ([r*(k + 1)] + [s*(k + 1)]) * (-x)^k, where [ ] = floor, r = (1+sqrt(5))/2, s = 1/r.

Original entry on oeis.org

1, 4, 11, 32, 94, 272, 786, 2272, 6564, 18962, 54780, 158254, 457174, 1320712, 3815354, 11022024, 31841080, 91984410, 265730044, 767656774, 2217652596, 6406486864, 18507440702, 53465396640, 154454021166, 446195972602, 1288997492332, 3723732703246
Offset: 0

Views

Author

Clark Kimberling, Aug 09 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Crossrefs

Cf. A078140.

Programs

  • Mathematica
    r = GoldenRatio; s = 1/GoldenRatio;
CoefficientList[Series[1/Sum[(Floor[r*(k + 1)] + Floor[s*(k + 1)]) (-x)^k, {k, 0, 1000}], {x, 0, 50}], x]

Formula

G.f.: 1/Sum_{k >= 0} ([r*(k + 1)] + [s*(k + 1)]) * (-x)^k, where [ ] = floor, r = (1+sqrt(5))/2, s = 1/r.
Previous Showing 31-39 of 39 results.

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