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).

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

0,1

Comments

See A195284 for definitions and a general discussion.

Links

Table of n, a(n) for n=0..101.

Example

Philo(ABC,I)=0.4629992818729451452524915088005478716250...

Mathematica

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 *)

Crossrefs

Cf. A195284.

Sequence in context: A171547 A114579 A021220 * A095196 A074828 A282028

Adjacent sequences: A195407 A195408 A195409 * A195411 A195412 A195413

Keyword

nonn,cons

Author

Clark Kimberling, Sep 17 2011

Extensions

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

Status

approved