A195500 -id:A195500 - cp's OEIS frontend

A195532 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(5).

8, 5, 84, 2400, 1691, 11480, 118455, 352692, 1401961, 1663145124, 1802526192, 15798984680, 297278169720, 1479041362764, 1551248530483, 42254295673488, 1445285680561323, 28154300465964144, 49087267967218280, 373205366478956820

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

Cf. A195500, A195533, A195534.

r = Sqrt[5]; z = 24;
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}]]  (* A195532, A195533 *)
Sqrt[a^2 + b^2] (* A195534 *)
(* by Peter J. C. Moses, Sep 02 2011 *)

A195535 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(6).

Original entry on oeis.org

5, 1020, 2247, 2633277900, 2640162496, 638843546289, 1396487515808, 6103353023795, 21860678072520, 82495605773137, 29466852345019792, 34041728665663572, 292320946605948260, 262936589866701605, 3964118460886936896

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

Cf. A195500, A195536, A195537.

r = Sqrt[6]; 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}]]  (* A195535, A195536 *)
Sqrt[a^2 + b^2] (* A195537 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195538 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

Original entry on oeis.org

5, 12, 145, 420, 4901, 14280, 166465, 485112, 5654885, 16479540, 192099601, 559819260, 6525731525, 19017375312, 221682772225, 646030941360, 7530688524101, 21946034630940, 255821727047185, 745519146510612, 8690408031080165

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

Conjecture: a(n) = 35*a(n-2) - 35*a(n-4) + a(n-6) with bisections A098602 and A076218. - R. J. Mathar, Sep 21 2011

Cf. A195500, A195539, A195540.

r = Sqrt[8]; z = 24;
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}]]  (* A195538, A195539 *)
Sqrt[a^2 + b^2] (* A195540 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195541 Denominators a(n) of Pythagorean approximations b(n)/a(n) to e.

Original entry on oeis.org

5, 12, 44, 51, 280, 949, 103488, 133416, 4142957, 81015132, 141119360, 2339121011, 22104171804, 658972588452, 461228244281, 3140753982224, 7467448353120, 49702513350325, 3912991025369532, 130254818350668557, 177768662787348689760

Offset: 1

Clark Kimberling, Sep 20 2011

nonn

See A195500 for a discussion and references.

Cf. A195500, A195542, A195543.

r = E; z = 23;
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}]]  (* A195541, A195542 *)
Sqrt[a^2 + b^2] (* A195543 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195544 Denominators a(n) of Pythagorean approximations b(n)/a(n) to Pi.

Original entry on oeis.org

12, 52, 315, 1044, 3296, 20919, 217620, 450296, 125510644, 138980066871, 289898680472, 3593636117787, 34812833117460, 1934468176818608, 1244148342635075, 86081645453428848, 8659539839551787053788, 138771143651019468539176

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

Cf. A195500, A195545, A195546.

r = Pi; z = 21;
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}]]  (* A195544, A195545 *)
Sqrt[a^2 + b^2] (* A195546 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195547 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/2.

Original entry on oeis.org

1, 4, 12, 15, 80, 208, 273, 1428, 3740, 4895, 25632, 67104, 87841, 459940, 1204140, 1576239, 8253296, 21607408, 28284465, 148099380, 387729212, 507544127, 2657535552, 6957518400, 9107509825, 47687540548, 124847601996, 163427632719, 855718194320, 2240299317520

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

a(n) is the numerator of the harmonic mean of F(n) and F(n+1), where F = A000045 (Fibonacci numbers). Example: 2*F(9)*F(10)/(F(9)+F(10)) = 2*34*55/(34+55) = 3740/89, therefore a(9) = 3740. - Francesco Daddi, Jul 04 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A195500, A195548, A195549, A131534.

r = 1/2; z = 30;
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}]]  (* A195547, A195548 *)
Sqrt[a^2 + b^2] (* A195549 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[Numerator[2 Fibonacci[n] Fibonacci[n+1] / ( Fibonacci[n] + Fibonacci[n+1])], {n, 1, 40}] (* Vincenzo Librandi, Jul 21 2018 *)

a(n) = 2*F(n)*F(n+1)/(2-((n+2)^2 mod 3)), where F(n)=Fibonacci(n). - Gary Detlefs, Oct 15 2011

Empirical G.f.: x*(1+4*x+12*x^2-2*x^3+12*x^4+4*x^5+x^6)/(1-17*x^3-17*x^6+x^9). - Colin Barker, Apr 15 2012

A195549 Hypotenuses of primitive Pythagorean triples in A195547 and A195548.

Original entry on oeis.org

1, 5, 13, 17, 89, 233, 305, 1597, 4181, 5473, 28657, 75025, 98209, 514229, 1346269, 1762289, 9227465, 24157817, 31622993, 165580141, 433494437, 567451585, 2971215073, 7778742049, 10182505537, 53316291173, 139583862445, 182717648081

Offset: 1

Clark Kimberling, Sep 20 2011

nonn

See A195500 for discussion and list of related sequences; see A195547 for Mathematica program.

Cf. A000045, A195500, A195547, A195548, A131534.

with(combinat):f:= n-> fibonacci(n):seq((f(n)^2+f(n+1)^2)/(2-((n+2)^2 mod 3)), n=1..25);

a(n) = (F(n)^2 + F(n+1)^2)/(2 - ((n+2)^2 mod 3)), F(n) = Fibonacci(n). - Gary Detlefs, Oct 14 2011
Conjectures from Colin Barker, Apr 08 2012: (Start)
a(n) = 18*a(n-3) - a(n-6).
G.f.: x*(1+5*x+13*x^2-x^3-x^4-x^5)/((1-3*x+x^2)*(1+3*x+8*x^2+3*x^3+x^4)). (End)
Conjecture: a(n) is the denominator of the reduced fraction (F(2*n+1)-2)/F(2*n+1), F(n) = Fibonacci(n). - Sébastien Labbé, May 06 2022

A195550 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 3/2.

Original entry on oeis.org

3, 60, 660, 3597, 78480, 856080, 4669203, 101866380, 1111191780, 6060621597, 132222483360, 1442326073760, 7866682164003, 171624681534300, 1872138132549300, 10210947388253997, 222768704409038640, 2430033853722917040

Offset: 1

Clark Kimberling, Sep 21 2011

See A195500 for a discussion and references.

Cf. A195500, A195551, A195552.

r = 3/2; z = 21;
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}]]  (* A195550, A195551 *)
Sqrt[a^2 + b^2] (* A195552 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195553 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 5/2.

Original entry on oeis.org

5, 260, 7020, 94635, 5103280, 137599280, 1855038645, 100034487540, 2697221086300, 36362467421275, 1960876019662560, 52870927596046560, 712777084536797285, 38437091637391006820, 1036375920040483589580, 13971856374727832955915

Offset: 1

Clark Kimberling, Sep 21 2011

See A195500 for a discussion and references.

Cf. A195500, A195554, A195555.

r = 5/2; 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}]]  (* A195553, A195554 *)
Sqrt[a^2 + b^2] (* A195555 *)
(* Peter J. C. Moses, Sep 02 2011 *)

A195556 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/3.

Original entry on oeis.org

1, 12, 24, 35, 468, 900, 1333, 17760, 34188, 50615, 674424, 1298232, 1922041, 25610340, 49298640, 72986939, 972518508, 1872050076, 2771581645, 36930092952, 71088604260, 105247115567, 1402371013680, 2699494911792, 3996618809905

Offset: 1

Clark Kimberling, Sep 21 2011

nonn

See A195500 for a discussion and references.

Cf. A195500, A195557, A195558.

r = 1/3; z = 27;
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}]]  (* A195556, A195557 *)
Sqrt[a^2 + b^2] (* A195558 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Conjecture: a(n) = 37*a(n-3) + 37*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011

Empirical g.f.: x*(x^6+12*x^5+24*x^4-2*x^3+24*x^2+12*x+1) / (x^9-37*x^6-37*x^3+1). - Colin Barker, Jun 04 2015

cp's OEIS Frontend

A195532 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(5).

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A195535 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(6).

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A195538 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

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A195541 Denominators a(n) of Pythagorean approximations b(n)/a(n) to e.

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A195544 Denominators a(n) of Pythagorean approximations b(n)/a(n) to Pi.

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A195547 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/2.

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A195549 Hypotenuses of primitive Pythagorean triples in A195547 and A195548.

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A195550 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 3/2.

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Mathematica

A195553 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 5/2.

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A195556 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/3.

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