cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A195579 Hypotenuses of primitive Pythagorean triples in A195577 and A195578.

Original entry on oeis.org

1, 5, 65, 349, 1189, 3709, 4549, 24425, 318365, 1709401, 5823721, 18166681, 22281001, 119633645, 1559351705, 8372645749, 28524584269, 88980399829, 109132338349, 585965568785, 7637704332725, 41009217169201, 139713407925841
Offset: 1

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

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

Crossrefs

Cf. A195500, A195577, A195578.

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

Views

Author

Clark Kimberling, Sep 20 2011

Keywords

Comments

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

Examples

			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...

Links

Crossrefs

Cf. A195501, A195502.

Programs

  • Maple
    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
  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Sep 21 2011

Keywords

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195578, A195579.

Programs

  • Mathematica
    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 *)

Extensions

Typos in crossrefs fixed by Colin Barker, Jun 04 2015

A195476 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(3),2).

Original entry on oeis.org

1, 2, 7, 2, 2, 2, 4, 6, 5, 6, 0, 9, 0, 3, 5, 2, 3, 3, 6, 6, 0, 8, 1, 4, 1, 9, 8, 1, 3, 6, 9, 2, 1, 8, 6, 0, 9, 5, 4, 9, 2, 0, 7, 5, 8, 8, 9, 4, 2, 5, 6, 3, 3, 0, 6, 9, 5, 6, 9, 4, 3, 5, 5, 8, 7, 1, 3, 6, 7, 4, 5, 3, 7, 4, 5, 2, 9, 4, 1, 8, 2, 3, 6, 0, 9, 7, 8, 6, 3, 3, 3, 5, 0, 1, 1, 8, 1, 8, 3, 5
Offset: 1

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(B)=1.272224656090352336608141981369218609549207...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195575 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B) A195576 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A195577 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)

A195477 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(3),2).

Original entry on oeis.org

9, 8, 8, 6, 5, 9, 2, 6, 2, 9, 8, 1, 9, 3, 8, 8, 4, 1, 7, 1, 3, 0, 9, 5, 8, 6, 3, 8, 8, 3, 8, 2, 5, 2, 4, 0, 3, 0, 6, 3, 3, 4, 0, 6, 3, 5, 4, 4, 3, 7, 8, 5, 1, 8, 2, 0, 8, 1, 0, 0, 4, 8, 2, 6, 1, 1, 8, 6, 8, 8, 8, 2, 0, 4, 0, 6, 8, 1, 2, 5, 5, 6, 8, 6, 4, 5, 6, 7, 3, 2, 1, 8, 6, 2, 9, 0, 6, 8, 2, 4
Offset: 0

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(C)=0.98865926298193884171309586388382524030...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195575 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B) A195576 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A195577 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)

A195478 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(3),2 right triangle ABC.

Original entry on oeis.org

6, 1, 3, 8, 4, 1, 7, 2, 5, 3, 9, 4, 1, 8, 6, 8, 1, 0, 6, 6, 0, 3, 6, 7, 2, 4, 6, 0, 0, 1, 3, 9, 4, 0, 2, 6, 6, 0, 7, 4, 8, 2, 7, 9, 6, 4, 8, 4, 2, 3, 9, 2, 9, 9, 9, 3, 8, 3, 0, 9, 0, 1, 7, 7, 7, 0, 9, 5, 7, 8, 7, 7, 1, 4, 1, 7, 5, 6, 4, 4, 4, 3, 6, 8, 4, 1, 2, 8, 9, 0, 4, 7, 2, 2, 2, 1, 4, 2, 9, 1
Offset: 0

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.61384172539418681066036724600139402660748...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195575 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B) A195576 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A195577 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)

A195475 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(3),2) and angles 30,60,90.

Original entry on oeis.org

6, 4, 3, 8, 4, 6, 3, 1, 3, 2, 9, 8, 7, 4, 3, 5, 3, 1, 5, 6, 9, 3, 7, 2, 1, 0, 7, 2, 1, 1, 8, 0, 9, 7, 2, 0, 6, 7, 5, 1, 9, 8, 1, 6, 0, 8, 2, 1, 8, 5, 8, 7, 2, 8, 7, 9, 9, 8, 8, 4, 7, 9, 2, 4, 7, 7, 6, 0, 4, 9, 3, 3, 7, 6, 7, 7, 9, 9, 8, 3, 9, 1, 9, 0, 0, 8, 7, 9, 2, 8, 3, 1, 3, 7, 8, 0, 4, 6, 5, 7
Offset: 0

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(A)=0.643846313298743531569372107211809720...

Crossrefs

Cf. A195304, A195476, A195477, A195478.

Programs

  • Mathematica
    a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195575 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B) A195576 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A195577 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)
Showing 1-7 of 7 results.

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