A195284 -id:A195284 - cp's OEIS frontend

A195407 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

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

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(A)=0.51252227236222537926354945581073551694...

Cf. A195284, A195408, A195409, A195410.

a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
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) A195407 *)
N[x2, 100]
RealDigits[%]    (* (B) A195408 *)
N[x3, 100]
RealDigits[%]    (* (C) A195409 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC,I) A195410 *)

A195408 Decimal expansion of shortest length, (B), of segment from side BC through incenter 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

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

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(B)=0.6119259581259097681148380144011707389...

Cf. A195284.

a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
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) A195407 *)
N[x2, 100]
RealDigits[%]    (* (B) A195408 *)
N[x3, 100]
RealDigits[%]    (* (C) A195409 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC,I) A195410 *)

A195409 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(C)=0.71278791738520123380160946972682714175360765866...

Cf. A195284.

a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
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) A195407 *)
N[x2, 100]
RealDigits[%]    (* (B) A195408 *)
N[x3, 100]
RealDigits[%]    (* (C) A195409 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC,I) A195410 *)

a(99) corrected by Georg Fischer, Jul 18 2021

A195410 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter 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

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

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.4629992818729451452524915088005478716250...

Cf. A195284.

a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
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) A195407 *)
N[x2, 100]
RealDigits[%]    (* (B) A195408 *)
N[x3, 100]
RealDigits[%]    (* (C) A195409 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC,I) A195410 *)

a(99) corrected by Georg Fischer, Jul 18 2021

A195286 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)=(5,12,13).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			(A)=4.0792156108742278640225792872182255...

Cf. A195284, A195288, A195289.

a = 5; b = 12; c = 13;
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) A195286 *)
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) A195288 *)
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) A010487 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)

A195288 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)=(5,12,13).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			(C)=4.80740170061865239082562835...

Cf. A195284.

a = 5; b = 12; c = 13;
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) A195286 *)
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) A195288 *)
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) A010487 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)

A195289 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.4847823853661753483351654180...

Cf. A195284.

a = 5; b = 12; c = 13;
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) A195286 *)
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) A195288 *)
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) A010487 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)

A195293 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)=(8,15,17).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			(A)=6.18465843842649082473211478396111...

Cf. A195284, A010524, A195296, A195297.

a = 8; b = 15; c = 17;
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) A195293 *)
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) A195296 *)
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), A195297 *)

A195296 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)=(8,15,17).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			(C)=6.99714227381436056504898345305469969182567800...

Cf. A195284.

a = 8; b = 15; c = 17;
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) A195293 *)
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) A195296 *)
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), A195297 *)

A195297 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of an 8,15,17 right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.54167705216198554206478076455685009252411270...

Cf. A195284.

a = 8; b = 15; c = 17;
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) A195293 *)
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) A195296 *)
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), A195297 *)

cp's OEIS Frontend

A195407 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

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A195408 Decimal expansion of shortest length, (B), of segment from side BC through incenter 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).

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Mathematica

A195409 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

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A195410 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of the right triangle ABC having sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

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Extensions

A195286 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)=(5,12,13).

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Mathematica

A195288 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)=(5,12,13).

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Mathematica

A195289 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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Mathematica

A195293 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)=(8,15,17).

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Mathematica

A195296 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)=(8,15,17).

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