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.

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A195298 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(28,45,53).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=20.800313969372909345992292832934379410...

Crossrefs

Cf. A195284, A195299, A195300, A195301.

Programs

  • Mathematica
    a = 28; b = 45; c = 53;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195298 *)
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, 1]
RealDigits[%, 10, 100] (* (B) A195299 *)
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] (* (C)=20*sqrt(2) *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)

A195299 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(28,45,53).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(C)=

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 28; b = 45; c = 53;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195298 *)
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, 1]
RealDigits[%, 10, 100] (* (B) A195299 *)
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] (* (C)=20*sqrt(2) *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)

A195300 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 28,45,53 right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.5711409786342621686192081085873951220...

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 28; b = 45; c = 53;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195298 *)
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, 1]
RealDigits[%, 10, 100] (* (B) A195299 *)
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] (* (C)=20*sqrt(2) *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)

A195301 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=0.63405067112442885068505288534396221319891000...

Links

Crossrefs

Cf. A195284, A195303, A195304.

Programs

  • Mathematica
    a = 1; b = 1; c = Sqrt[2];
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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, 1]
RealDigits[%, 10, 100] (* (A) A195301 *)
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] (* (B)=(A) *)
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, 1]
RealDigits[%, 10, 100] (* (C) A163960 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Philo(ABC,I), A195303 *)

A195290 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(7,24,25).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=6.0609152673132644948643802466...

Crossrefs

Cf. A195284, A195292.

Programs

  • Mathematica
    a = 7; b = 24; c = 25;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195290 *)
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, 1]
RealDigits[%, 10, 100] (* (B)=7.5 *)
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] (* (C) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)

A195292 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 7,24,25 right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.39368208288485419263704486771198536...

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 7; b = 24; c = 25;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195290 *)
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, 1]
RealDigits[%, 10, 100] (* (B)=7.5 *)
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] (* (C) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)

A195303 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,1,sqrt(2) right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion. This constant is the maximum of Philo(ABC,I) over all triangles ABC.

Examples

			Philo(ABC,I)=0.614058971032212611546384892539385408260...

Links

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 1; b = 1; c = Sqrt[2];
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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, 1]
RealDigits[%, 10, 100] (* (A) A195301 *)
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] (* (B)=(A) *)
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, 1]
RealDigits[%, 10, 100] (* (C) A163960 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Philo(ABC,I), A195303 *)
  • PARI
    (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))) \\ Michel Marcus, Jul 27 2018

Formula

Equals (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))).

A195348 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(3),2) and vertex angles of degree measure 30,60,90.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 17 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=0.7578747639260239988121867474270095303467925401944...
(A)=(4*sqrt(6-3*sqrt(3)))/(3+sqrt(3))
(B)=2-(2/3)sqrt(3)
(C)=sqrt(6)-sqrt(2)

Crossrefs

Cf. A195284, A093821, A120683, A195380.

Programs

  • Mathematica
    a = 1; b = Sqrt[3]; c = 2;
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%] (* (A) A195348 *)
N[x2, 100]
RealDigits[%] (* (B) A093821 *)
N[x3, 100]
RealDigits[%] (* (C) A120683 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195380 *)

A195380 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,sqrt(3),sqrt(1) right triangle ABC (angles 30, 60, 90).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 17 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.55757017691709380372112914604292318...

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 1; b = Sqrt[3]; c = 2;
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%] (* (A) A195348 *)
N[x2, 100]
RealDigits[%] (* (B) A093821 *)
N[x3, 100]
RealDigits[%] (* (C) A120683 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195380 *)

A197034 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197008 and A195284.
A root of the polynomial x^3-7*x^2+18*x-20. - R. J. Mathar, Nov 08 2022

Examples

			length of Philo line:   2.60819402496101...; see A197035
endpoint on x axis:   (3.47797, 0)
endpoint on line y=x: (2.35321, 2.35321)

Crossrefs

Cf. A197032, A197035, A197008, A195284.

Programs

  • Mathematica
    f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] :=  h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3
m = 1; h = 3; k = 1;  (* slope m; point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197034 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)], N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197035 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 3.5}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]

Extensions

Last digit removed (representation truncated, not rounded up). - R. J. Mathar, Nov 08 2022
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