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

0,1

Comments

See A195284 for definitions and a general discussion.

Links

Table of n, a(n) for n=0..100.

Index entries for algebraic numbers, degree 8.

Example

Philo(ABC,I)=0.4280818058125219350252671517036980901568443655...

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 *)

Prog

(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

Crossrefs

Cf. A195284.

Sequence in context: A375447 A201414 A199814 * A200693 A348681 A241298

Adjacent sequences: A195344 A195345 A195346 * A195348 A195349 A195350

Keyword

nonn,cons

Author

Clark Kimberling, Sep 16 2011

Extensions

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

Status

approved