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 *)
A097727
Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n >= 0.
Original entry on oeis.org
1, 101, 10301, 1050601, 107151001, 10928351501, 1114584702101, 113676711262801, 11593909964103601, 1182465139627304501, 120599850332020955501, 12300002268726510156601, 1254479631559772015017801, 127944622416828019021659101, 13049097006884898168194210501
Offset: 0
(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 =-1.
-
a:=[1,101];; for n in [3..20] do a[n]:=102*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
-
I:=[1,101]; [n le 2 select I[n] else 102*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
-
LinearRecurrence[{102,-1},{1,101},20] (* Harvey P. Dale, Apr 12 2014 *)
CoefficientList[Series[(1-x)/(1-102x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Apr 13 2014 *)
-
my(x='x+O('x^20)); Vec((1-x)/(1-102*x+x^2)) \\ G. C. Greubel, Aug 01 2019
-
((1-x)/(1-102*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
A195623
Numerators of Pythagorean approximations to 5.
Original entry on oeis.org
99, 10101, 1030199, 105070201, 10716130299, 1092940220301, 111469186340399, 11368764066500401, 1159502465596700499, 118257882726796950501, 12061144535667692250599, 1230118484755377812610601, 125460024300512869194030699, 12795692360167557279978520701, 1305035160712790329688615080799
Offset: 1
-
I:=[99,10101,1030199]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 16 2023
-
Table[(5*LucasL[2*n+1,10] +2*(-1)^n)/52, {n,40}] (* G. C. Greubel, Feb 16 2023 *)
-
Vec(-x*(x^2-102*x-99) / ((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015
-
A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(1/26)*(25*A097726(n) + (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 16 2023
Showing 1-3 of 3 results.
Comments