A195389 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

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

Offset

0,1

Comments

See A195284 for definitions and a general discussion.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Index entries for algebraic numbers, degree 16

Example

Philo(ABC,I)=0.5738949427486823068594102114264402286939808...

Mathematica

a = Sqrt[2]; b = Sqrt[5]; c = Sqrt[7];
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) A195386 *)
N[x2, 100]
RealDigits[%]   (* (A) A195387 *)
N[x3, 100]
RealDigits[%]   (* (A) A195388 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC, I) A195389 *)

Prog

(PARI) (10*sqrt(14*sqrt(14)+98) + 12*sqrt(10*sqrt(14)+70) + 10*sqrt(14*sqrt(35)+98) + 14*sqrt(10*sqrt(35)+70) + 45*sqrt(sqrt(35)+7) + 9*sqrt(35*sqrt(35)+245) + 140*sqrt(2) + 24*sqrt(70) + 70*sqrt(sqrt(14)+7) + 12*sqrt(35*sqrt(14)+245) + 90*sqrt(7) + 108*sqrt(5)) / (299*sqrt(10)+253*sqrt(14)+169*sqrt(35)+1001) \\ Charles R Greathouse IV, Apr 07 2026
(PARI) polrootsreal(390625*x^16 - 637125000*x^14 + 8232000000*x^13 + 77985110000*x^12 - 1176223440000*x^11 + 991255739600*x^10 + 15668406051200*x^9 - 3406075705144*x^8 - 41165096000000*x^7 + 11569416728800*x^6 + 24413234406400*x^5 - 3902307662400*x^4 - 4050391744000*x^3 - 158212536000*x^2 + 126929600000*x + 11978090000)[10] \\ Charles R Greathouse IV, Apr 07 2026

Crossrefs

Cf. A195284, A195386, A195387, A195388.

Sequence in context: A152081 A264736 A200620 * A345653 A091663 A011316

Adjacent sequences: A195386 A195387 A195388 * A195390 A195391 A195392

Keyword

nonn,cons

Author

Clark Kimberling, Sep 17 2011

Status

approved