A195503 -id:A195503 - cp's OEIS frontend

A195531 Hypotenuses of Pythagorean triples in A195499 and A195503.

5, 17, 65, 241, 901, 3361, 12545, 46817, 174725, 652081, 2433601, 9082321, 33895685, 126500417, 472105985, 1761923521, 6575588101, 24540428881, 91586127425, 341804080817, 1275630195845, 4760716702561, 17767236614401, 66308229755041

Offset: 1

Clark Kimberling, Sep 20 2011

nonn

See A195500 for discussion and list of related sequences; see A195499 for Mathematica program.

Essentially the same as A120893. - R. J. Mathar, Sep 21 2011

Cf. A195500, A195499, A195503.

Conjecture: a(n) = 3*a(n-1)+3*a(n-2)-a(n-3). G.f.: x*(5+2*x-x^2)/((1+x)*(1-4*x+x^2)). - Colin Barker, Apr 08 2012

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

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 *)

A195499 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).

Original entry on oeis.org

3, 8, 33, 120, 451, 1680, 6273, 23408, 87363, 326040, 1216801, 4541160, 16947843, 63250208, 236052993, 880961760, 3287794051, 12270214440, 45793063713, 170902040408, 637815097923, 2380358351280, 8883618307201, 33154114877520

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

Apparently a(n) = A120892(n+1) for 1 <= n <= 24. - Georg Fischer, Oct 24 2018

			From the Pythagorean triples (3,4,5), (8,15,17),(33,56,65), (120,209,241), (451,780,901), read the first five best approximating fractions b(n)/a(n):
4/3, 15/8, 56/33, 209/120, 780/451.

Cf. A120892, A195500, A195503, A195531.

r = Sqrt[3]; z = 25;
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}]]  (* A195499, A195503 *)
Sqrt[a^2 + b^2] (* A195531 *)
(* by Peter J. C. Moses, Sep 02 2011 *)

Empirical G.f.: x*(3-x)/(1-3*x-3*x^2+x^3). - Colin Barker, Jan 04 2012

A195501 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(2).

Original entry on oeis.org

4, 325, 435, 7480, 769189, 998691, 1760400, 72332699, 107770201, 162402622743, 150321171634588, 314779738565193, 4979439027990791, 25240412071733925, 26320772661145332, 150784475760058387, 381611630092964177, 506075333191877232

Offset: 1

Clark Kimberling, Sep 20 2011

nonn

See A195500 for discussion, Mathematica program, and list of related sequences.

Cf. A195501, A195503.

A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15.

Original entry on oeis.org

4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244

Offset: 0

Simon Plouffe

nonn

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A010925, A001353, A079935, A195503.

/* By definition: */ [n le 2 select 11*n-7 else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..22]]; // Bruno Berselli, Apr 16 2012

a[0] = 4; a[1] = 15; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 24}] (* Michael De Vlieger, Jul 27 2016 *)

pisotE(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
  a
}
pisotE(50, 4, 15) \\ Colin Barker, Jul 27 2016

@cached_function
def A010905(n):
    if n==0: return 4
    elif n==1: return 15
    else: return 4*A010905(n-1) - A010905(n-2)
[A010905(n) for n in range(30)] # G. C. Greubel, Dec 13 2018

a(n) = 4*a(n-1) - a(n-2) for n>=2. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016
This was conjectured by Colin Barker, Apr 16 2012, and implies the G.f.: (4-x)/(1-4*x+x^2) and the formula a(n) = ((1+sqrt(3))^(2*n+4)-(1-sqrt(3))^(2*n+4))/(2^(n+3)*sqrt(3)).
Partial sums of A079935. - Erin Pearse, Dec 13 2018

Edited by N. J. A. Sloane, Jul 26 2016 and Sep 09 2016

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A195531 Hypotenuses of Pythagorean triples in A195499 and A195503.

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

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A195499 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).

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A195501 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(2).

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A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15.

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