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 *)
A078988
Chebyshev sequence with Diophantine property.
Original entry on oeis.org
1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585, 29220420180900779828304449, 1928104896615996323709378049
Offset: 0
(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.
- Colin Barker, Table of n, a(n) for n = 0..549
- A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
- Tanya Khovanova, Recursive Sequences
- Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (66,-1).
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a:=[1,65];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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I:=[1, 65]; [n le 2 select I[n] else 66*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
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CoefficientList[Series[(1-x)/(1-66x+x^2), {x,0,20}], x] (* Michael De Vlieger, Apr 15 2019 *)
LinearRecurrence[{66,-1}, {1,65}, 21] (* G. C. Greubel, Aug 01 2019 *)
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Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015
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((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
A195620
Numerators of Pythagorean approximations to 4.
Original entry on oeis.org
63, 4161, 274559, 18116737, 1195430079, 78880268481, 5204902289663, 343444670849281, 22662143373762879, 1495358017997500737, 98670967044461285759, 6510788466916447359361, 429613367849441064432063, 28347971489596193805156801
Offset: 1
-
I:=[63,4161,274559]; [n le 3 select I[n] else 65*Self(n-1) +65*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023
-
LinearRecurrence[{65,65,-1}, {63,4161,274559}, 40] (* G. C. Greubel, Feb 15 2023 *)
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Vec(x*(63+66*x-x^2)/((1+x)*(1-66*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 03 2015
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A078989=BinaryRecurrenceSequence(66, -1, 1, 67)
[(16*A078989(n) + (-1)^n)/17 for n in range(1, 41)] # G. C. Greubel, Feb 15 2023
Showing 1-3 of 3 results.
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