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

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

Offset

1,1

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

Links

Table of n, a(n) for n=1..18.

Ron Knott, Pythagorean Angles

Peter Shiu, The shapes and sizes of Pythagorean triangles, The Mathematical Gazette 67, no. 439 (March 1983) 33-38.

Example

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...

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 *)

Crossrefs

Cf. A195501, A195502.

Sequence in context: A254157 A131493 A228871 * A099426 A332123 A100201

Adjacent sequences: A195497 A195498 A195499 * A195501 A195502 A195503

Keyword

nonn,frac

Author

Clark Kimberling, Sep 20 2011

Status

approved