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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A195559 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2/3.

Original entry on oeis.org

1, 4, 91, 240, 840, 989, 5396, 117719, 312120, 1089720, 1284121, 7003804, 152799571, 405130920, 1414456320, 1666787669, 9090932396, 198333725039, 525859622640, 1835963213040, 2163489110641, 11800023246004, 257437022301451
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195560, A195561.

Programs

  • Mathematica
    r = 2/3; z = 25;
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}]]  (* A195559, A195560 *)
Sqrt[a^2 + b^2] (* A195561 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195561 Hypotenuses of primitive Pythagorean triples in A195559 and A195560.

Original entry on oeis.org

1, 5, 109, 289, 1009, 1189, 6485, 141481, 375121, 1309681, 1543321, 8417525, 183642229, 486906769, 1699964929, 2003229469, 10925940965, 238367471761, 632004611041, 2206553168161, 2600190307441, 14181862955045, 309400794703549
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195559 for Mathematica program.

Crossrefs

Cf. A195500, A195559, A195560.

Formula

Empirical g.f.: -x*(x^9+x^8+x^7+5*x^6+109*x^5-1009*x^4-289*x^3-109*x^2-5*x-1) / (x^10-1298*x^5+1). - Colin Barker, Jun 04 2015

A195562 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/4.

Original entry on oeis.org

1, 24, 40, 63, 1600, 2624, 4161, 105560, 173160, 274559, 6965376, 11425920, 18116737, 459609240, 753937576, 1195430079, 30327244480, 49748454080, 78880268481, 2001138526424, 3282644031720, 5204902289663, 132044815499520
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195563, A195564.

Programs

  • Mathematica
    Remove["Global`*"];
r = 1/4; z = 26;
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}]]  (* A195562, A195563 *)
Sqrt[a^2 + b^2] (* A195564 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Conjecture: a(n) = 65*a(n-3) + 65*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011
Empirical g.f.: x*(x^6+24*x^5+40*x^4-2*x^3+40*x^2+24*x+1) / (x^9-65*x^6-65*x^3+1). - Colin Barker, Jun 04 2015

A195565 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 5/4.

Original entry on oeis.org

4, 3, 204, 280, 2280, 12596, 12797, 831996, 1149440, 9341440, 51622404, 52441603, 3409523404, 4710402840, 38281220840, 211548594996, 214905676797, 13972226073596, 19303229690880, 156876433658880, 866926090675204, 880683411072003
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195566, A195567.

Programs

  • Mathematica
    r = 5/4; z = 26;
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}]]  (* A195565, A195566 *)
Sqrt[a^2 + b^2] (* A195567 *)
(* by Peter J. C. Moses, Sep 02 2011 *)

A195568 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 7/4.

Original entry on oeis.org

3, 8, 120, 637, 2176, 30848, 164483, 561288, 7958776, 42435837, 144810240, 2053333248, 10948281603, 37360480520, 529752019320, 2824614217597, 9638859164032, 136673967651200, 728739519858563, 2486788303839624, 35261353901990392
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195569, A195570.

Programs

  • Mathematica
    r = 7/4; z = 26;
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}]]  (* A195568, A195569 *)
Sqrt[a^2 + b^2] (* A195570 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Empirical g.f.: x*(3*x^6+8*x^5+120*x^4-134*x^3+120*x^2+8*x+3) / (x^9-257*x^6-257*x^3+1). - Colin Barker, Jun 04 2015

A195571 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/5.

Original entry on oeis.org

1, 40, 60, 99, 4100, 6100, 10101, 418140, 622160, 1030199, 42646200, 63454200, 105070201, 4349494240, 6471706260, 10716130299, 443605766300, 660050584300, 1092940220301, 45243438668340, 67318687892360, 111469186340399, 4614387138404400
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195572, A195573.

Programs

  • Mathematica
    r = 1/5; z = 26;
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}]]  (* A195571, A195572 *)
Sqrt[a^2 + b^2] (* A195573 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Conjecture: a(n) = 101*a(n-3) + 101*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011
Empirical g.f.: x*(x^6+40*x^5+60*x^4-2*x^3+60*x^2+40*x+1) / (x^9-101*x^6-101*x^3+1). - Colin Barker, Jun 04 2015

A195574 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2/5.

Original entry on oeis.org

1, 12, 651, 5720, 12480, 17549, 236588, 12758199, 112130200, 244626200, 343998201, 4637596612, 250086218851, 2197976167920, 4795162766680, 6743052715749, 90906168553188, 4902190049156399, 43084728731444400, 93994780307828400
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195575, A195576.

Programs

  • Mathematica
    r = 2/5; z = 26;
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}]]  (* A195574, A195575 *)
Sqrt[a^2 + b^2] (* A195576 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195580 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 4/5.

Original entry on oeis.org

4, 600, 2600, 15996, 2460800, 10652800, 65552004, 10084355800, 43655173800, 268632095996, 41325687609600, 178898891577600, 1100854263840004, 169352657739783000, 733127614029833000, 4511300504584239996, 694007150091943126400
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195611, A195612.

Programs

  • Mathematica
    r = 4/5; z = 18;
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}]]  (* A195580, A195611 *)
Sqrt[a^2 + b^2] (* A195612 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195617 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3.

Original entry on oeis.org

35, 1333, 50615, 1922041, 72986939, 2771581645, 105247115567, 3996618809905, 151766267660819, 5763121552301221, 218846852719785575, 8310417281799550633, 315577009855663138475, 11983615957233399711421, 455061829365013525895519
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. A085447, A097315, A195500, A195616.

Programs

  • Magma
    I:=[35, 1333, 50615]; [n le 3 select I[n] else 37*Self(n-1) +37*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 13 2023
  • Mathematica
    Table[(3*LucasL[2*n+1,6] +2*(-1)^n)/20, {n, 40}] (* G. C. Greubel, Feb 13 2023 *)
  • PARI
    Vec(-x*(x^2-38*x-35)/((x+1)*(x^2-38*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015
  • SageMath
    A085447=BinaryRecurrenceSequence(6,1,2,6)
[(3*A085447(2*n+1) + 2*(-1)^n)/20 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

Formula

From Colin Barker, Jun 04 2015: (Start)
a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3).
G.f.: x*(35+38*x-x^2) / ((1+x)*(1-38*x+x^2)). (End)
a(n) = (1/20)*(3*A085447(2*n+1) + 2*(-1)^n). - G. C. Greubel, Feb 13 2023

A195619 Denominators of Pythagorean approximations to 4.

Original entry on oeis.org

16, 1040, 68640, 4529184, 298857520, 19720067120, 1301225572416, 85861167712320, 5665535843440720, 373839504499375184, 24667741761115321440, 1627697116729111839840, 107403341962360266108016, 7086992872399048451289200
Offset: 1

Views

Author

Clark Kimberling, Sep 22 2011

Keywords

Comments

See A195500 for a discussion and references.

Links

Crossrefs

Cf. A078988, A078989, A195500, A195620.

Programs

  • Magma
    I:=[16, 1040, 68640]; [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 13 2023
  • Mathematica
    r = 4; 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}]]  (* A195619, A195620 *)
Sqrt[a^2 + b^2] (* A078988 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(LucasL[2*n+1,8] - 8*(-1)^n)/34, {n,40}] (* G. C. Greubel, Feb 13 2023 *)
LinearRecurrence[{65,65,-1},{16,1040,68640},20] (* Harvey P. Dale, May 01 2023 *)
  • PARI
    Vec(16*x/((x+1)*(x^2-66*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015
  • SageMath
    A078989=BinaryRecurrenceSequence(66,-1,1,67)
[4*(A078989(n) - (-1)^n)/17 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

Formula

From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: 16*x/((1+x)*(1-66*x+x^2)). (End)
a(n) = ((4+sqrt(17))^(2*n+1) + (4-sqrt(17))^(2*n+1) - 8*(-1)^n)/34. - Colin Barker, Mar 03 2016
a(n) = 4*(A078989(n) - (-1)^n)/17. - G. C. Greubel, Feb 13 2023
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