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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A195496 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(B)=1.017153446754804466256798187816606336...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = b - 1; b = GoldenRatio; 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) A195495 *)
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) A195496 *)
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] (* (C) A195497 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)

A195497 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(C)=0.862968792141037434136010433016173...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = b - 1; b = GoldenRatio; 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) A195495 *)
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) A195496 *)
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] (* (C) A195497 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)

A195498 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the right triangle ABC having sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.575915236513482373618787369187419918767270...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = b - 1; b = GoldenRatio; 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) A195495 *)
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) A195496 *)
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] (* (C) A195497 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)

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 *)

A195305 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)=(3,4,5).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 18 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(B)=3.288554185145030064182848108896351414361583823030...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 3; b = 4; 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) A195304 *)
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) A195305 *)
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]   (* (C) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)

A195306 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)=(3,4,5).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 18 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(C)=2.3725916749567493080750985299403201500573613270...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 3; b = 4; 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) A195304 *)
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) A195305 *)
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]   (* (C) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)

A195411 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 3,4,5 right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 18 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.629787202200915115584317820207624290124920...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 3; b = 4; 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) A195304 *)
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) A195305 *)
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]   (* (C) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)

A195433 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)=(1,1,sqrt(2)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 18 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(A)=0.6147572272333906215933192480911...

Links

Crossrefs

Cf. A195304, A195434, A195435, A195436.

Programs

  • Mathematica
    a = 1; b = 1; 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) A195433 *)
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)=(2/3)sqrt(2); -1+A179587 *)
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] (* (C) A195433 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)

A195436 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 18 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.636258821061838308391049464710...

Links

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = 1; b = 1; 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) A195433 *)
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)=sqrt(8/9), -1+A179587  *)
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] (* (C) A195433 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)

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 *)
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