A195620 -id:A195620 - 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 *)

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

Wolfdieter Lang, Jan 10 2003

Bisection (even part) of A041025.
(4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).
Starting with a(1), hypotenuses of primitive Pythagorean triples in A195619 and A195620. - Clark Kimberling, Sep 22 2011

			(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).

Row 66 of array A094954.
Cf. A097316 for S(n, 66).
Row 4 of array A188647.

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

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

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 *)

Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015

((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

G.f.: (1-x)/(1-66*x+x^2).
a(n) = T(2*n+1, sqrt(17))/sqrt(17) = ((-1)^n)*S(2*n, 8*i) = S(n, 66) - S(n-1, 66) with i^2=-1 and T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
a(n) = A041025(2*n).
a(n) = 66*a(n-1) - a(n-2) for n>1 ; a(0)=1, a(1)=65. - Philippe Deléham, Nov 18 2008

A195619 Denominators of Pythagorean approximations to 4.

Original entry on oeis.org

16, 1040, 68640, 4529184, 298857520, 19720067120, 1301225572416, 85861167712320, 5665535843440720, 373839504499375184, 24667741761115321440, 1627697116729111839840, 107403341962360266108016, 7086992872399048451289200

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for a discussion and references.

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

Cf. A078988, A078989, A195500, A195620.

I:=[16, 1040, 68640]; [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 13 2023

r = 4; z = 20;
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}]]  (* A195619, A195620 *)
Sqrt[a^2 + b^2] (* A078988 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(LucasL[2*n+1,8] - 8*(-1)^n)/34, {n,40}] (* G. C. Greubel, Feb 13 2023 *)
LinearRecurrence[{65,65,-1},{16,1040,68640},20] (* Harvey P. Dale, May 01 2023 *)

Vec(16*x/((x+1)*(x^2-66*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

A078989=BinaryRecurrenceSequence(66,-1,1,67)
[4*(A078989(n) - (-1)^n)/17 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: 16*x/((1+x)*(1-66*x+x^2)). (End)
a(n) = ((4+sqrt(17))^(2*n+1) + (4-sqrt(17))^(2*n+1) - 8*(-1)^n)/34. - Colin Barker, Mar 03 2016
a(n) = 4*(A078989(n) - (-1)^n)/17. - G. C. Greubel, Feb 13 2023

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

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A078988 Chebyshev sequence with Diophantine property.

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A195619 Denominators of Pythagorean approximations to 4.

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