A195500 -id:A195500 - cp's OEIS frontend

A195502 Hypotenuses of primitive Pythagorean triples in A195500 and A195501.

5, 397, 533, 9161, 942061, 1223141, 2156041, 88589101, 131991001, 198901779305, 184105084021037, 385524870425705, 6098542411938841, 30913065236666477, 32236231327801693, 184672513372600885, 467376886819742065, 619813168864541257

Offset: 1

Clark Kimberling, Sep 20 2011

nonn

See A195500 for discussion, Mathematica program, and guide to related sequences.

Cf. A195500, A195501.

A195577 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 3/5.

Original entry on oeis.org

1, 4, 56, 299, 1020, 3180, 3901, 20944, 272996, 1465799, 4993800, 15577800, 19105801, 102585004, 1337133056, 7179484499, 24459629220, 76300063380, 93580208101, 502461329944, 6549277433996, 35165113611599, 119803258923600

Offset: 1

Clark Kimberling, Sep 21 2011

See A195500 for a discussion and references.

Cf. A195500, A195578, A195579.

r = 3/5; z = 26;
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}]]  (* A195577, A195578 *)
Sqrt[a^2 + b^2] (* A195579 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Typos in crossrefs fixed by Colin Barker, Jun 04 2015

A195575 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 2/5.

Original entry on oeis.org

0, 5, 260, 2289, 4991, 7020, 94635, 5103280, 44852079, 97850481, 137599280, 1855038645, 100034487540, 879190467169, 1918065106671, 2697221086300, 36362467421275, 1960876019662560, 17233891492577759, 37597912123131361

Offset: 1

Clark Kimberling, Sep 21 2011

See A195500 for discussion and list of related sequences; see A195574 for Mathematica program.

Cf. A195500, A195574, A195576.

A195576 Hypotenuses of primitive Pythagorean triples in A195574 and A195575.

Original entry on oeis.org

1, 13, 701, 6161, 13441, 18901, 254813, 13741001, 120767921, 263470481, 370497401, 4994844413, 269351100901, 2367292781281, 5164548355121, 7262490035501, 97908939928813, 5279820266120401, 46403672977902241

Offset: 1

Clark Kimberling, Sep 21 2011

nonn

See A195500 for discussion and list of related sequences; see A195574 for Mathematica program.

Cf. A195500, A195574, A195575.

A195578 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3/5.

Original entry on oeis.org

0, 3, 33, 180, 611, 1909, 2340, 12567, 163797, 879480, 2996279, 9346681, 11463480, 61551003, 802279833, 4307690700, 14675777531, 45780038029, 56148124860, 301476797967, 3929566460397, 21099068166960, 71881955354159, 224230616915761

Offset: 1

Clark Kimberling, Sep 21 2011

See A195500 for discussion and list of related sequences; see A195577 for Mathematica program.

Cf. A195500, A195577, A195579.

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

A195503 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).

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

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for discussion and list of related sequences; see A195499 for Mathematica program.
Essentially the same as A001353, A106707, A125905 and likely also A010905. - R. J. Mathar, Sep 21 2011
If started 1, 4, 15, 56, ...., denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(1/3). - Clark Kimberling, Sep 22 2011

Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A195500, A195499, A195531.

a(n) = A001353(n+1). - R. J. Mathar, Sep 21 2011

Empirical g.f.: x*(4-x)/(1-4*x+x^2). - Colin Barker, Jan 04 2012

A195564 Hypotenuses of primitive Pythagorean triples in A195561 and A195562.

Original entry on oeis.org

1, 25, 41, 65, 1649, 2705, 4289, 108809, 178489, 283009, 7179745, 11777569, 18674305, 473754361, 777141065, 1232221121, 31260608081, 51279532721, 81307919681, 2062726378985, 3383672018521, 5365090477825, 136108680404929

Offset: 1

Clark Kimberling, Sep 21 2011

nonn

See A195500 for discussion and list of related sequences; see A195562 for Mathematica program.

Cf. A195500, A195562, A195563.

Empirical g.f.: -x*(x^5+x^4+x^3-41*x^2-25*x-1) / (x^6-66*x^3+1). - Colin Barker, Jun 04 2015

A195616 Denominators of Pythagorean approximations to 3.

Original entry on oeis.org

12, 444, 16872, 640680, 24328980, 923860548, 35082371856, 1332206269968, 50588755886940, 1921040517433740, 72948950906595192, 2770139093933183544, 105192336618554379492, 3994538652411133237140, 151687276455004508631840

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for a discussion and references.

Colin Barker, Table of n, a(n) for n = 1..633
Index entries for linear recurrences with constant coefficients, signature (37,37,-1).

Cf. A085447, A097314, A097315, A195500, A195617.

I:=[12, 444, 16872]; [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

r = 3; 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}]]  (* A195616, A195617 *)
Sqrt[a^2 + b^2] (* A097315 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(1/20)*(LucasL[2*n+1,6] -6*(-1)^n), {n,40}] (* G. C. Greubel, Feb 13 2023 *)

Vec(12*x/((1+x)*(1-38*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 04 2015

A085447=BinaryRecurrenceSequence(6,1,2,6)
[(A085447(2*n+1) - 6*(-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.: 12*x / ((1+x)*(1-38*x+x^2)). (End)
From G. C. Greubel, Feb 13 2023: (Start)
a(n) = (3/10)*(A097314(n) + (-1)^n).
a(n) = (1/20)*(A085447(2*n+1) - 6*(-1)^n). (End)

A195625 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(1/2).

Original entry on oeis.org

1, 4, 756, 435, 7480, 1760400, 178342500, 107770201, 77071637784, 85438755160, 162402622743, 150321171634588, 314779738565193, 1395140327976600, 3899078438579384, 26320772661145332, 506075333191877232, 6916169937061302541

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for a discussion and references.

Cf. A195500, A195626, A195627.

r = Sqrt[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}]]  (* A195625, A195626 *)
Sqrt[a^2 + b^2] (* A195627 *)
(* Peter J. C. Moses, Sep 02 2011 *)

cp's OEIS Frontend

A195502 Hypotenuses of primitive Pythagorean triples in A195500 and A195501.

Views

Author

Keywords

Comments

Crossrefs

A195577 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 3/5.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A195575 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 2/5.

Views

Author

Keywords

Comments

Crossrefs

A195576 Hypotenuses of primitive Pythagorean triples in A195574 and A195575.

Views

Author

Keywords

Comments

Crossrefs

A195578 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3/5.

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A195503 Numerators b(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A195564 Hypotenuses of primitive Pythagorean triples in A195561 and A195562.

Views

Author

Keywords

Comments

Crossrefs

Formula

A195616 Denominators of Pythagorean approximations to 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A195625 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(1/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica