A195616 -id:A195616 - cp's OEIS frontend

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

Clark Kimberling, Sep 20 2011

For each positive real number r, there is a sequence (a(n),b(n),c(n)) of primitive Pythagorean triples such that the limit of b(n)/a(n) is r and
|r-b(n+1)/a(n+1)| < |r-b(n)/a(n)|.  Peter Shiu showed how to find (a(n),b(n)) from the continued fraction for r, and Peter J. C. Moses incorporated Shiu's method in the Mathematica program shown below.
Examples:
r...........a(n)..........b(n)..........c(n)
sqrt(2).....A195500.......A195501.......A195502
sqrt(3).....A195499.......A195503.......A195531
sqrt(5).....A195532.......A195533.......A195534
sqrt(6).....A195535.......A195536.......A195537
sqrt(8).....A195538.......A195539.......A195540
sqrt(12)....A195680.......A195681.......A195682
e...........A195541.......A195542.......A195543
pi..........A195544.......A195545.......A195546
tau.........A195687.......A195688.......A195689
1...........A046727.......A084159.......A001653
2...........A195614.......A195615.......A007805
3...........A195616.......A195617.......A097315
4...........A195619.......A195620.......A078988
5...........A195622.......A195623.......A097727
1/2.........A195547.......A195548.......A195549
3/2.........A195550.......A195551.......A195552
5/2.........A195553.......A195554.......A195555
1/3.........A195556.......A195557.......A195558
2/3.........A195559.......A195560.......A195561
1/4.........A195562.......A195563.......A195564
5/4.........A195565.......A195566.......A195567
7/4.........A195568.......A195569.......A195570
1/5.........A195571.......A195572.......A195573
2/5.........A195574.......A195575.......A195576
3/5.........A195577.......A195578.......A195579
4/5.........A195580.......A195611.......A195612
sqrt(1/2)...A195625.......A195626.......A195627
sqrt(1/3)...{1}+A195503...{0}+A195499...{1}+A195531
sqrt(2/3)...A195631.......A195632.......A195633
sqrt(3/4)...A195634.......A195635.......A195636

			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...

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

Cf. A195501, A195502.

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 *)

A097315 Pell equation solutions (3b(n))^2 - 10a(n)^2 = -1 with b(n) = A097314(n), n >= 0.

Original entry on oeis.org

1, 37, 1405, 53353, 2026009, 76934989, 2921503573, 110940200785, 4212806126257, 159975692596981, 6074863512559021, 230684837784645817, 8759948972303982025, 332647376109766671133, 12631840343198829521029, 479677285665445755127969, 18215105014943739865341793, 691694313282196669127860165

Offset: 0

Wolfdieter Lang, Aug 31 2004

Hypotenuses of primitive Pythagorean triples in A195616 and A195617. - Clark Kimberling, Sep 22 2011

			(x,y) = (3,1), (117,37), (4443,1405), ... give the positive integer solutions to x^2 - 10*y^2 = -1.
G.f. = 1 + 37*x + 1405*x^2 + 53353*x^3 + ... - _Michael Somos_, Feb 24 2023

Indranil Ghosh, Table of n, a(n) for n = 0..631
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 (38,-1).

Cf. A078987 (partial sums), A097314, A195616, A195617, A049310, A053120, A005668.
Row 3 of array A188647.
Cf. A221874.
Cf. similar sequences listed in A238379.

a:=[1,37];; for n in [3..20] do a[n]:=38*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1, 37]; [n le 2 select I[n] else 38*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019

CoefficientList[Series[(1-x)/(1-38x+x^2), {x,0,20}], x] (* Michael De Vlieger, Feb 04 2017 *)
LinearRecurrence[{38,-1}, {1,37}, 21] (* G. C. Greubel, Aug 01 2019 *)

Vec((1-x)/(1-38*x+x^2) + O(x^20)) \\ Michel Marcus, Jun 04 2015

from itertools import islice
def A097315_gen(): # generator of terms
    x, y = 30, 10
    while True:
        yield y//10
        x, y = x*19+y*60, x*6+y*19
A097315_list = list(islice(A097315_gen(),20)) # Chai Wah Wu, Apr 24 2025

((1-x)/(1-38*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 38) - S(n-1, 38) = T(2*n+1, sqrt(10))/sqrt(10), with Chebyshev polynomials of the second and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 6*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-38*x+x^2).
a(n) = 38*a(n-1) - a(n-2) for n > 1. - Philippe Deléham, Nov 18 2008
a(n) = sqrt(2+(19-6*sqrt(10))^(1+2*n) + (19+6*sqrt(10))^(1+2*n))/(2*sqrt(10)). - Gerry Martens, Jun 04 2015
a(n) = A078987(n) - A078987(n-1). - R. J. Mathar, Dec 05 2015
a(n) = A005668(2*n+1). - Michael Somos, Feb 24 2023
E.g.f.: exp(19*x)*(10*cosh(6*sqrt(10)*x) + 3*sqrt(10)*sinh(6*sqrt(10)*x))/10. - Stefano Spezia, Apr 24 2025

Typo in recurrence formula corrected by Laurent Bonaventure (bonave(AT)free.fr), Oct 03 2010

More terms added by Indranil Ghosh, Feb 04 2017

A195617 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3.

Original entry on oeis.org

35, 1333, 50615, 1922041, 72986939, 2771581645, 105247115567, 3996618809905, 151766267660819, 5763121552301221, 218846852719785575, 8310417281799550633, 315577009855663138475, 11983615957233399711421, 455061829365013525895519

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for discussion and list of related sequences; see A195616 for Mathematica program.

Colin Barker, Table of n, a(n) for n = 1..632
Index entries for linear recurrences with constant coefficients, signature (37,37,-1).

Cf. A085447, A097315, A195500, A195616.

I:=[35, 1333, 50615]; [n le 3 select I[n] else 37*Self(n-1) +37*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 13 2023

Table[(3*LucasL[2*n+1,6] +2*(-1)^n)/20, {n, 40}] (* G. C. Greubel, Feb 13 2023 *)

Vec(-x*(x^2-38*x-35)/((x+1)*(x^2-38*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015

A085447=BinaryRecurrenceSequence(6,1,2,6)
[(3*A085447(2*n+1) + 2*(-1)^n)/20 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

From Colin Barker, Jun 04 2015: (Start)
a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3).
G.f.: x*(35+38*x-x^2) / ((1+x)*(1-38*x+x^2)). (End)
a(n) = (1/20)*(3*A085447(2*n+1) + 2*(-1)^n). - G. C. Greubel, Feb 13 2023

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

Clark Kimberling, Sep 22 2011

See A195500 for discussion and list of related sequences; see A195616 for Mathematica program.

Colin Barker, Table of n, a(n) for n = 1..549
Index entries for linear recurrences with constant coefficients, signature (65,65,-1).

Cf. A078989, A195500, A195619, A195620.

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 *)

Vec(x*(63+66*x-x^2)/((1+x)*(1-66*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 03 2015

A078989=BinaryRecurrenceSequence(66, -1, 1, 67)
[(16*A078989(n) + (-1)^n)/17 for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: x*(63+66*x-x^2) / ((1+x)*(1-66*x+x^2)). (End)
a(n) = ((-1)^n - 2*(-4+sqrt(17))*(33+8*sqrt(17))^(-n) + 2*(4+sqrt(17))*(33+8*sqrt(17))^n)/17. - Colin Barker, Mar 03 2016
a(n) = (1/17)*(A078989(n) + (-1)^n) - [n=0]. - G. C. Greubel, Feb 15 2023

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A195500 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(2).

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A097315 Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n) = A097314(n), n >= 0.

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A195617 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3.

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A097315 Pell equation solutions (3b(n))^2 - 10a(n)^2 = -1 with b(n) = A097314(n), n >= 0.