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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A289922 Coefficients of 1/([1+r] - [1+2r]x + [1+3r]x^2 - ...), where [ ] = floor and r = 19/21.

Original entry on oeis.org

1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 2, -1, -5, -4, -1, 0, 0, 0, 0, 1, 1, -6, -15, -14, -6, -1, 0, 0, 0, 1, 0, -10, -21, -18, -7, -1, 0, 0, 0, 1, -1, -13, -20, -3, 18, 18, 7, 1, 0, 1, -2, -15, -13, 29
Offset: 0

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Links

Crossrefs

Cf. A078140 (includes guide to related sequences), A289921, A289923.

Programs

  • Mathematica
    z = 2000; r = 19/21;
CoefficientList[Series[1/Sum[Floor[1 + (k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];
  • PARI
    Vec((1 + x)^2*(1 - x + x^2)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6)*(1 + x - x^3 - x^4 + x^6 - x^8 - x^9 + x^11 + x^12) / (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19 + x^20 + x^21) + O(x^100)) \\ Colin Barker, Jul 21 2017

Formula

G.f.: 1/([1+r] - [1+2r]x + [1+3r]x^2 - ...), where [ ] = floor and r = 19/21.
G.f.: (1 + x)^2*(1 - x + x^2)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6)*(1 + x - x^3 - x^4 + x^6 - x^8 - x^9 + x^11 + x^12) / (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19 + x^20 + x^21). - Colin Barker, Jul 20 2017

A289923 Limiting sequence of coefficients of 1/([1+r] - [1+2r]x + [1+3r]x^2 - ...), where [ ] = floor and r approaches 19/21 from the left.

Original entry on oeis.org

1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 2, 7, 9, 5, 1, 0, 0, 0, 0, 0, 3, 12, 19, 15, 6, 1, 0, 0, 0, 0, 5, 22, 40, 39, 22, 7, 1, 0, 0, 0, 8, 39, 81, 94, 67, 30, 8, 1, 0, 0, 13, 69, 160, 214, 183, 104, 39, 9, 1, 0, 21, 121, 310, 468, 464
Offset: 0

Views

Author

Clark Kimberling, Jul 18 2017

Keywords

Comments

Conjecture: all the terms are nonnegative.

Links

Crossrefs

Cf. A078140 (includes guide to related sequences), A289921, A289922.

Programs

  • Mathematica
    z = 2000; r = 19/21-10^(-9);
CoefficientList[Series[1/Sum[Floor[1 + (k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];

Formula

G.f.: 1/([1+r] - [1+2r]x + [1+3r]x^2 - ...), where [ ] = floor and r = 19/21-10^(-9).
G.f.: (1 + x)^2*(1 - x + x^2)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6)*(1 + x - x^3 - x^4 + x^6 - x^8 - x^9 + x^11 + x^12) / (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19). - Colin Barker, Jul 20 2017

A288229 Coefficients of 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = Pi/2 and [ ] = floor.

Original entry on oeis.org

1, 3, 5, 9, 18, 36, 72, 144, 287, 570, 1132, 2250, 4473, 8892, 17676, 35137, 69847, 138845, 276002, 548649, 1090629, 2168001, 4309649, 8566912, 17029689, 33852374, 67293256, 133768530, 265911039, 528589801, 1050754338, 2088736250, 4152082903, 8253695235
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 = Pi/2;
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 = Pi/2 and [ ] = floor.

A288230 Coefficients of 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = sqrt(5/2) and [ ] = floor.

Original entry on oeis.org

1, 3, 5, 9, 18, 36, 71, 138, 268, 522, 1017, 1980, 3853, 7498, 14594, 28406, 55287, 107604, 209428, 407608, 793325, 1544042, 3005154, 5848902, 11383662, 22155913, 43121842, 83927627, 163347533, 317921733, 618768013, 1204302235, 2343921860, 4561952576
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[5/2];
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(5/2) and [ ] = floor.

A288231 Coefficients of 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = 4^(1/3) and [ ] = floor.

Original entry on oeis.org

1, 3, 5, 9, 18, 36, 71, 138, 268, 522, 1017, 1981, 3859, 7517, 14642, 28521, 55557, 108223, 210814, 410654, 799931, 1558224, 3035341, 5912689, 11517614, 22435718, 43703622, 85132404, 165833537, 323035186, 629255898, 1225758065, 2387713549, 4651142959
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[5/2];
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 = 4^(1/3) and [ ] = floor.

A288232 Coefficients in the expansion of 1/([r]-[2*r]*x+[3*r]*x^2-...); []=floor, r=3*e/5.

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, 4343663, 7662423, 13516866, 23844368, 42062554, 74200268, 130892661, 230900629, 407319256, 718529778
Offset: 0

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Crossrefs

Cf. A078140 (includes guide to related sequences).

Programs

  • Mathematica
    r = 3E/5;
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 = 3*e/5 and [ ] = floor.

A288233 Coefficients in the expansion of 1/([r]-[2*r]*x+[3*r]*x^2-...); [ ]=floor, r=sqrt(8/3).

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 91, 160, 281, 494, 871, 1537, 2711, 4782, 8437, 14885, 26258, 46319, 81706, 144126, 254229, 448442, 791021, 1395308, 2461230, 4341448, 7658035, 13508286, 23827758, 42030652, 74139404, 130777206, 230682689, 406909610, 717762700
Offset: 0

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Examples

			G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = sqrt(8/3) and [ ] = floor.

Crossrefs

Cf. A078140 (includes guide to related sequences).

Programs

  • Mathematica
    r = Sqrt[8/3];
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]

A288234 Coefficients in the expansion of 1/([r]-[2*r]*x+[3*r]*x^2-...); [ ]=floor, r=-1+sqrt(7).

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 91, 160, 281, 493, 865, 1518, 2664, 4675, 8204, 14398, 25271, 44356, 77853, 136647, 239844, 420976, 738898, 1296915, 2276349, 3995455, 7012834, 12308945, 21604693, 37920614, 66558359, 116823399, 205048721, 359902025, 631700929
Offset: 0

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Comments

Conjecture: the sequence is strictly increasing.

Examples

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

Crossrefs

Cf. A078140 (includes guide to related sequences).

Programs

  • Mathematica
    r = -1 + Sqrt[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]

A288238 Decimal expansion of the limiting ratio of consecutive terms of A288135.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Examples

			1.99998855479481596316339920436515588837259520599...

Crossrefs

Cf. A288135, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = Sqrt[7/3];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288135 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A288238 *)

A288239 Decimal expansion of the limiting ratio of consecutive terms of A288229.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2017

Keywords

Examples

			1.9878445175009702132085952971222...

Crossrefs

Cf. A288229, A078140 (includes guide to related constants).

Programs

  • Mathematica
    z = 2000; r = Pi/2;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288229 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A288239 *)
Previous Showing 11-20 of 39 results. Next

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