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%I #38 Apr 21 2019 04:38:44
%S 3,228,308,5289,543900,706180,1244791,51146940,76205040,114835995824,
%T 106293119818725,222582887719576,3520995103197240,17847666535865852,
%U 18611596834765355,106620725307595884,269840171418387336,357849299891217865
%N Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(2).
%C 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
%C |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.
%C Examples:
%C r...........a(n)..........b(n)..........c(n)
%C sqrt(2).....A195500.......A195501.......A195502
%C sqrt(3).....A195499.......A195503.......A195531
%C sqrt(5).....A195532.......A195533.......A195534
%C sqrt(6).....A195535.......A195536.......A195537
%C sqrt(8).....A195538.......A195539.......A195540
%C sqrt(12)....A195680.......A195681.......A195682
%C e...........A195541.......A195542.......A195543
%C pi..........A195544.......A195545.......A195546
%C tau.........A195687.......A195688.......A195689
%C 1...........A046727.......A084159.......A001653
%C 2...........A195614.......A195615.......A007805
%C 3...........A195616.......A195617.......A097315
%C 4...........A195619.......A195620.......A078988
%C 5...........A195622.......A195623.......A097727
%C 1/2.........A195547.......A195548.......A195549
%C 3/2.........A195550.......A195551.......A195552
%C 5/2.........A195553.......A195554.......A195555
%C 1/3.........A195556.......A195557.......A195558
%C 2/3.........A195559.......A195560.......A195561
%C 1/4.........A195562.......A195563.......A195564
%C 5/4.........A195565.......A195566.......A195567
%C 7/4.........A195568.......A195569.......A195570
%C 1/5.........A195571.......A195572.......A195573
%C 2/5.........A195574.......A195575.......A195576
%C 3/5.........A195577.......A195578.......A195579
%C 4/5.........A195580.......A195611.......A195612
%C sqrt(1/2)...A195625.......A195626.......A195627
%C sqrt(1/3)...{1}+A195503...{0}+A195499...{1}+A195531
%C sqrt(2/3)...A195631.......A195632.......A195633
%C sqrt(3/4)...A195634.......A195635.......A195636
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#pythangs">Pythagorean Angles</a>
%H Peter Shiu, <a href="http://www.jstor.org/stable/3617358">The shapes and sizes of Pythagorean triangles</a>, The Mathematical Gazette 67, no. 439 (March 1983) 33-38.
%e 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):
%e (3,4,5); |r - b(1)/a(1)| = 0.08...
%e (228,325,397); |r - b(2)/a(2)| = 0.011...
%e (308,435,533); |r - b(3)/a(3)| = 0.0018...
%e (5289,7480,9161); |r - b(4)/a(4)| = 0.000042...
%e (543900,769189,942061); |r - b(5)/a(5)| = 0.0000003...
%p Shiu := proc(r,n)
%p t := r+sqrt(1+r^2) ;
%p cf := numtheory[cfrac](t,n+1) ;
%p mn := numtheory[nthconver](cf,n) ;
%p (mn-1/mn)/2 ;
%p end proc:
%p A195500 := proc(n)
%p Shiu(sqrt(2),n) ;
%p denom(%) ;
%p end proc: # _R. J. Mathar_, Sep 21 2011
%t r = Sqrt[2]; z = 18;
%t p[{f_, n_}] := (#1[[2]]/#1[[
%t 1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t 2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t Array[FromContinuedFraction[
%t ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t p[{r, z}]] (* A195500, A195501 *)
%t Sqrt[a^2 + b^2] (* A195502 *)
%Y Cf. A195501, A195502.
%K nonn,frac
%O 1,1
%A _Clark Kimberling_, Sep 20 2011
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