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 A195500 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(2). 80
 3, 228, 308, 5289, 543900, 706180, 1244791, 51146940, 76205040, 114835995824, 106293119818725, 222582887719576, 3520995103197240, 17847666535865852, 18611596834765355, 106620725307595884, 269840171418387336, 357849299891217865 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified December 2 20:59 EST 2022. Contains 358510 sequences. (Running on oeis4.)