A195304 -id:A195304 - cp's OEIS frontend

A195486 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			Philo(ABC,G)=.62033227265302583805568683720607688648361...

Cf. A195304.

a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)
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) A195484 *)
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) A195485 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)

A195487 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)=(sqrt(7),3,4).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(A)=1.648040734595518812330740700...

Cf. A195304, A195488, A195489, A195490.

a = Sqrt[7]; b = 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) A195487 *)
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) A195488 *)
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) A195489  *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)

A195488 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)=(sqrt(7),3,4).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(B)=2.659684722763015782869315876506123197220...

Cf. A195304.

a = Sqrt[7]; b = 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) A195487 *)
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) A195488 *)
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) A195489  *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)

A195489 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)=(sqrt(7),3,4).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(C)=1.817363600575517623762638911647...

Cf. A195304.

a = Sqrt[7]; b = 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) A195487 *)
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) A195488 *)
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) A195489  *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)

A195490 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(7),3,4 right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			Philo(ABC,G)=0.635003833336237352470212190369350359...

Cf. A195304.

a = Sqrt[7]; b = 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) A195487 *)
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) A195488 *)
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) A195489  *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)

A195491 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(A)=0.62958106138771604404549587568854069...

Cf. A195304, A195492, A195493, A195494.

a = 1; b = Sqrt[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) A195491 *)
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) A195492 *)
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) A195493 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)

A195492 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(B)=1.0684730171986100357539820259956...

Index entries for algebraic numbers, degree 8

Cf. A195304.

a = 1; b = Sqrt[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) A195491 *)
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) A195492 *)
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) A195493 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)

polrootsreal(3464*x^8 - 16028*x^7 + 40544*x^6 + 47079*x^5 - 64883*x^4 + 50115*x^3 - 43008*x^2 - 60658*x + 32292)[4] \\ Charles R Greathouse IV, Nov 26 2024

A195493 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(C)=0.759310778373734956811842690497767...

Cf. A195304.

a = 1; b = Sqrt[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) A195491 *)
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) A195492 *)
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) A195493 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)

A195494 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the right triangle ABC having sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			Philo(ABC,G)=0.631704620416679682980614446416476083...

Cf. A195304.

a = 1; b = Sqrt[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) A195491 *)
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) A195492 *)
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) A195493 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)

A195495 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(A)=0.40517782972057177788030739498051458468832393740...

Cf. A195304, A195496, A195497, A195498.

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 *)

cp's OEIS Frontend

A195486 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

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A195487 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)=(sqrt(7),3,4).

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A195488 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)=(sqrt(7),3,4).

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Mathematica

A195489 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)=(sqrt(7),3,4).

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Mathematica

A195490 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(7),3,4 right triangle ABC.

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Mathematica

A195491 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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Mathematica

A195492 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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PARI

A195493 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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Author

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Examples

Crossrefs

Programs

Mathematica

A195494 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the right triangle ABC having sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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