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
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
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...
-
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
-
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
Showing 1-3 of 3 results.
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