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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A195620 Numerators of Pythagorean approximations to 4.

Original entry on oeis.org

63, 4161, 274559, 18116737, 1195430079, 78880268481, 5204902289663, 343444670849281, 22662143373762879, 1495358017997500737, 98670967044461285759, 6510788466916447359361, 429613367849441064432063, 28347971489596193805156801
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195616 for Mathematica program.

Links

Crossrefs

Cf. A078989, A195500, A195619, A195620.

Programs

  • Magma
    I:=[63,4161,274559]; [n le 3 select I[n] else 65*Self(n-1) +65*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023
  • Mathematica
    LinearRecurrence[{65,65,-1}, {63,4161,274559}, 40] (* G. C. Greubel, Feb 15 2023 *)
  • PARI
    Vec(x*(63+66*x-x^2)/((1+x)*(1-66*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 03 2015
  • SageMath
    A078989=BinaryRecurrenceSequence(66, -1, 1, 67)
[(16*A078989(n) + (-1)^n)/17 for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

Formula

From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: x*(63+66*x-x^2) / ((1+x)*(1-66*x+x^2)). (End)
a(n) = ((-1)^n - 2*(-4+sqrt(17))*(33+8*sqrt(17))^(-n) + 2*(4+sqrt(17))*(33+8*sqrt(17))^n)/17. - Colin Barker, Mar 03 2016
a(n) = (1/17)*(A078989(n) + (-1)^n) - [n=0]. - G. C. Greubel, Feb 15 2023

A195622 Denominators of Pythagorean approximations to 5.

Original entry on oeis.org

20, 2020, 206040, 21014040, 2143226060, 218588044060, 22293837268080, 2273752813300080, 231900493119340100, 23651576545359390100, 2412228907133538450120, 246023696951075562522120, 25092004860102573838806140, 2559138472033511455995704140
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for a discussion and references.

Links

Crossrefs

Cf. A097726, A195500, A195623.

Programs

  • Magma
    I:=[20,2020,206040]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023
  • Mathematica
    r = 5; z = 20;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195622, A195623 *)
Sqrt[a^2 + b^2] (* A097727 *)
(* by Peter J. C. Moses, Sep 02 2011 *)
LinearRecurrence[{101,101,-1},{20,2020,206040},20] (* Harvey P. Dale, Oct 17 2021 *)
  • PARI
    Vec(20*x/((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015
  • SageMath
    A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(5/26)*(A097726(n) - (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

Formula

From Colin Barker, Jun 03 2015: (Start)
a(n) = 101*a(n-1) + 101*a(n-2) - a(n-3).
G.f.: 20*x/((1+x)*(1-102*x+x^2)). (End)
a(n) = (5/26)*(A097726(n) - (-1)^n). - G. C. Greubel, Feb 15 2023

A195623 Numerators of Pythagorean approximations to 5.

Original entry on oeis.org

99, 10101, 1030199, 105070201, 10716130299, 1092940220301, 111469186340399, 11368764066500401, 1159502465596700499, 118257882726796950501, 12061144535667692250599, 1230118484755377812610601, 125460024300512869194030699, 12795692360167557279978520701, 1305035160712790329688615080799
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195622 for Mathematica program.

Links

Crossrefs

Cf. A097726, A097727, A195500, A195622.

Programs

  • Magma
    I:=[99,10101,1030199]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 16 2023
  • Mathematica
    Table[(5*LucasL[2*n+1,10] +2*(-1)^n)/52, {n,40}] (* G. C. Greubel, Feb 16 2023 *)
  • PARI
    Vec(-x*(x^2-102*x-99) / ((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015
  • SageMath
    A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(1/26)*(25*A097726(n) + (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 16 2023

Formula

From Colin Barker, Jun 03 2015: (Start)
a(n) = 101*a(n-1) + 101*a(n-2) - a(n-3).
G.f.: x*(99+102*x-x^2)/((1+x)*(1-102*x+x^2)). (End)
a(n) = (1/26)*(25*A097726(n) + (-1)^n). - G. C. Greubel, Feb 16 2023
E.g.f.: (5*exp(51*x)*(5*cosh(10*sqrt(26)*x) + sqrt(26)*sinh(10*sqrt(26)*x)) + exp(-x) - 26)/26. - Stefano Spezia, Aug 05 2024

A195631 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(2/3).

Original entry on oeis.org

4, 171, 3304, 36456, 193028, 198629, 64044140, 3209176272, 2089963197, 14714161192, 151173075361, 450458512764, 1490895165780, 4767682119956876, 19409457907183648, 293100434264580753, 818944253254104320, 19191303984466705047
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195632, A195633.

Programs

  • Mathematica
    r = Sqrt[2/3]; z = 28;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195631, A195632 *)
Sqrt[a^2 + b^2] (* A195633 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195634 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(3/4).

Original entry on oeis.org

4, 55, 1120, 2997, 35460, 3140676, 1921787, 32412552, 58579212, 441025780, 410535015, 77779347592, 654610870027, 2025218645520, 7961709199049, 29306172663680, 88433963478036, 109778426942667, 2900499582545112, 4716082204442140
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195635, A195636.

Programs

  • Mathematica
    r = Sqrt[3/4]; z = 28;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195634, A195635 *)
Sqrt[a^2 + b^2] (* A195636 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195680 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(12).

Original entry on oeis.org

7, 2772, 5945, 26144, 285621, 257076560, 2386970016, 103850955649, 254621037540, 3060691213613, 29733959304728, 62837775720000, 89511043811115, 453985767379732, 1567652657852541, 35830073055128140, 22926879590846577132
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195681, A195682.

Programs

  • Mathematica
    r = Sqrt[12]; z = 24;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195680, A195681 *)
Sqrt[a^2 + b^2] (* A195682 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195681 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(12).

Original entry on oeis.org

24, 9605, 20592, 90567, 989420, 890539329, 8268706687, 359750263200, 882033147389, 10602545376516, 103001456451945, 217676440363319, 310075351438748, 1572652830029685, 5430508104041980, 124119013940773131, 79421040620720449885
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195680 for Mathematica program.

Crossrefs

Cf. A195500, A195680, A195682.

A195682 Hypotenuses of primitive Pythagorean triples in A195680 and A195681.

Original entry on oeis.org

25, 9997, 21433, 94265, 1029821, 926902721, 8606342785, 374439945601, 918049206661, 11035479109045, 107207314895753, 226564822394569, 322736658181277, 1636868962618493, 5652252040004741, 129187165603885181, 82664039951186181157
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195680 for Mathematica program.

Crossrefs

Cf. A195500, A195680, A195681.

A195687 Denominators a(n) of Pythagorean approximations b(n)/a(n) to (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

3, 8, 28, 1863, 4400, 433008, 262353, 352207108, 379920428, 18418959496, 91011249895, 978117768540, 11516765628956, 1219780690817560, 708294344602810604, 25852535312829023356, 229222230912132985022679
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195681, A195682.

Programs

  • Mathematica
    r = GoldenRatio; z = 24;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[p[{r, z}]]  (* A195687, A195688 *)
Sqrt[a^2 + b^2] (* A195689 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195501 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(2).

Original entry on oeis.org

4, 325, 435, 7480, 769189, 998691, 1760400, 72332699, 107770201, 162402622743, 150321171634588, 314779738565193, 4979439027990791, 25240412071733925, 26320772661145332, 150784475760058387, 381611630092964177, 506075333191877232
Offset: 1

Views

Author

Clark Kimberling, Sep 20 2011

Keywords

Comments

See A195500 for discussion, Mathematica program, and list of related sequences.

Crossrefs

Cf. A195501, A195503.
Previous Showing 31-40 of 79 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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