A195347 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,3,sqrt(10) right triangle ABC.

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

Offset: 0

Author

Clark Kimberling, Sep 16 2011

Keywords

• nonn
• cons

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.4280818058125219350252671517036980901568443655...
		

Links

• Index entries for algebraic numbers, degree 8.

Crossrefs

Cf. A195284.

Programs

Mathematica

a = 1; b = 3; c = Sqrt[10]; 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[%] (* A195344 *)
N[x2, 100]
RealDigits[%] (* A195345 *)
N[x3, 100]
RealDigits[%] (* A195346 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195347 *)

PARI

polrootsreal(6561*x^8 - 42107688*x^6 + 495305280*x^5 + 39826979224*x^4 - 60652800*x^3 - 4964068512*x^2 - 900806400*x - 44270064)[6] \\ Charles R Greathouse IV, Feb 11 2025

Extensions

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