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
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
- Ron Knott, Pythagorean Angles
- Peter Shiu, The shapes and sizes of Pythagorean triangles, The Mathematical Gazette 67, no. 439 (March 1983) 33-38.
Programs
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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 *)
Comments