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.

A195540 Hypotenuses of primitive Pythagorean triples in A195538 and A195539.

Original entry on oeis.org

13, 37, 433, 1261, 14701, 42841, 499393, 1455337, 16964653, 49438621, 576298801, 1679457781, 19577194573, 57052125937, 665048316673, 1938092824081, 22592065572301, 65838103892821, 767465181141553, 2236557439531837, 26071224093240493
Offset: 1

Views

Author

Clark Kimberling, Sep 20 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195538 for Mathematica program.

Crossrefs

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...
		

Crossrefs

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 *)

A195539 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

Original entry on oeis.org

12, 35, 408, 1189, 13860, 40391, 470832, 1372105, 15994428, 46611179, 543339720, 1583407981, 18457556052, 53789260175, 627013566048, 1827251437969, 21300003689580, 62072759630771, 723573111879672, 2108646576008245, 24580185800219268
Offset: 1

Views

Author

Clark Kimberling, Sep 20 2011

Keywords

Comments

See A195500 for discussion and list of related sequences; see A195538 for Mathematica program.
Conjecture: a(n) = +34*a(n-2) -a(n-4) with a(2n+1) = A000129(4n+4), a(2n)=A046176(n+1). - R. J. Mathar, Sep 21 2011

Crossrefs

Showing 1-3 of 3 results.