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 *)
A195680
Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(12).
Original entry on oeis.org
7, 2772, 5945, 26144, 285621, 257076560, 2386970016, 103850955649, 254621037540, 3060691213613, 29733959304728, 62837775720000, 89511043811115, 453985767379732, 1567652657852541, 35830073055128140, 22926879590846577132
Offset: 1
-
r = Sqrt[12]; z = 24;
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}]] (* A195680, A195681 *)
Sqrt[a^2 + b^2] (* A195682 *)
(* Peter J. C. Moses, Sep 02 2011 *)
A195681
Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(12).
Original entry on oeis.org
24, 9605, 20592, 90567, 989420, 890539329, 8268706687, 359750263200, 882033147389, 10602545376516, 103001456451945, 217676440363319, 310075351438748, 1572652830029685, 5430508104041980, 124119013940773131, 79421040620720449885
Offset: 1
A195687
Denominators a(n) of Pythagorean approximations b(n)/a(n) to (1+sqrt(5))/2 (the golden ratio).
Original entry on oeis.org
3, 8, 28, 1863, 4400, 433008, 262353, 352207108, 379920428, 18418959496, 91011249895, 978117768540, 11516765628956, 1219780690817560, 708294344602810604, 25852535312829023356, 229222230912132985022679
Offset: 1
-
r = GoldenRatio; z = 24;
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}]] (* A195687, A195688 *)
Sqrt[a^2 + b^2] (* A195689 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Showing 1-4 of 4 results.
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