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.

Showing 1-3 of 3 results.

A195500 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(2).

Original entry on oeis.org

3, 228, 308, 5289, 543900, 706180, 1244791, 51146940, 76205040, 114835995824, 106293119818725, 222582887719576, 3520995103197240, 17847666535865852, 18611596834765355, 106620725307595884, 269840171418387336, 357849299891217865
Offset: 1

Views

Author

Clark Kimberling, Sep 20 2011

Keywords

Comments

For each positive real number r, there is a sequence (a(n),b(n),c(n)) of primitive Pythagorean triples such that the limit of b(n)/a(n) is r and
|r-b(n+1)/a(n+1)| < |r-b(n)/a(n)|. Peter Shiu showed how to find (a(n),b(n)) from the continued fraction for r, and Peter J. C. Moses incorporated Shiu's method in the Mathematica program shown below.
Examples:
r...........a(n)..........b(n)..........c(n)
sqrt(2).....A195500.......A195501.......A195502
sqrt(3).....A195499.......A195503.......A195531
sqrt(5).....A195532.......A195533.......A195534
sqrt(6).....A195535.......A195536.......A195537
sqrt(8).....A195538.......A195539.......A195540
sqrt(12)....A195680.......A195681.......A195682
e...........A195541.......A195542.......A195543
pi..........A195544.......A195545.......A195546
tau.........A195687.......A195688.......A195689
1...........A046727.......A084159.......A001653
2...........A195614.......A195615.......A007805
3...........A195616.......A195617.......A097315
4...........A195619.......A195620.......A078988
5...........A195622.......A195623.......A097727
1/2.........A195547.......A195548.......A195549
3/2.........A195550.......A195551.......A195552
5/2.........A195553.......A195554.......A195555
1/3.........A195556.......A195557.......A195558
2/3.........A195559.......A195560.......A195561
1/4.........A195562.......A195563.......A195564
5/4.........A195565.......A195566.......A195567
7/4.........A195568.......A195569.......A195570
1/5.........A195571.......A195572.......A195573
2/5.........A195574.......A195575.......A195576
3/5.........A195577.......A195578.......A195579
4/5.........A195580.......A195611.......A195612
sqrt(1/2)...A195625.......A195626.......A195627
sqrt(1/3)...{1}+A195503...{0}+A195499...{1}+A195531
sqrt(2/3)...A195631.......A195632.......A195633
sqrt(3/4)...A195634.......A195635.......A195636

Examples

			For r=sqrt(2), the first five fractions b(n)/a(n) can be read from the following five primitive Pythagorean triples (a(n), b(n), c(n)) = (A195500, A195501, A195502):
(3,4,5); |r - b(1)/a(1)| = 0.08...
(228,325,397); |r - b(2)/a(2)| = 0.011...
(308,435,533); |r - b(3)/a(3)| = 0.0018...
(5289,7480,9161); |r - b(4)/a(4)| = 0.000042...
(543900,769189,942061); |r - b(5)/a(5)| = 0.0000003...

Links

Crossrefs

Cf. A195501, A195502.

Programs

  • Maple
    Shiu := proc(r,n)
        t := r+sqrt(1+r^2) ;
        cf := numtheory[cfrac](t,n+1) ;
        mn := numtheory[nthconver](cf,n) ;
        (mn-1/mn)/2 ;
end proc:
A195500 := proc(n)
        Shiu(sqrt(2),n) ;
        denom(%) ;
end proc: # R. J. Mathar, Sep 21 2011
  • Mathematica
    r = Sqrt[2]; 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}]]  (* A195500, A195501 *)
Sqrt[a^2 + b^2] (* A195502 *)

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

A195572 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 1/5.

Original entry on oeis.org

0, 9, 11, 20, 819, 1221, 2020, 83629, 124431, 206040, 8529239, 12690841, 21014040, 869898849, 1294341251, 2143226060, 88721153259, 132010116861, 218588044060, 9048687733669, 13463737578471, 22293837268080, 922877427680879
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195571 for Mathematica program.
Conjecture: satisfies a linear recurrence having signature (0, 0, 101, 0, 0, 101, 0, 0, -1). - Harvey P. Dale, Jun 13 2025

Crossrefs

Cf. A195500, A195571, A195573.

Formula

Empirical g.f.: -x^2*(x^7-x^6-110*x^4+90*x^3-20*x^2-11*x-9) / (x^9-101*x^6-101*x^3+1). - Colin Barker, Jun 04 2015
Showing 1-3 of 3 results.

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