A195398 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a sqrt(3),sqrt(5),sqrt(8) right triangle ABC.

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

Offset

0,1

Comments

See A195284 for definitions and a general discussion.

Links

Table of n, a(n) for n=0..102.

Index entries for algebraic numbers, degree 20

Example

Philo(ABC,I)=0.6011263969176532516541263787772610806686403999...

Mathematica

a = Sqrt[3]; b = Sqrt[5]; c = Sqrt[8];
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) A195395 *)
N[x2, 100]
RealDigits[%]    (* (B) A195396 *)
N[x3, 100]
RealDigits[%]    (* (C) A195397 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC, I) A195398 *)

Prog

(PARI) (120*sqrt(2*sqrt(6)+8) + 78*sqrt(5*sqrt(6)+20) + 120*sqrt(2*sqrt(10)+8) + 110*sqrt(3*sqrt(10)+12) + 96*sqrt(5*sqrt(10)+20) + 44*sqrt(30*sqrt(10)+120) + 480*sqrt(2) + 312*sqrt(5) + 160*sqrt(3*sqrt(6)+12) + 52*sqrt(30*sqrt(6)+120) + 440*sqrt(3) + 143*sqrt(30)) / (783*sqrt(15)+1242*sqrt(6)+986*sqrt(10)+3128) \\ Charles R Greathouse IV, Apr 07 2026
(PARI) polrootsreal(2562890625*x^20 + 2562890625*x^19 - 3196824468750*x^18 + 40284004640625*x^17 + 108673929500625*x^16 - 2853495781380000*x^15 + 7824300936483600*x^14 + 19159513716826800*x^13 - 94000973321476784*x^12 + 69241450475782816*x^11 - 5832154083770368*x^10 + 112141084996107616*x^9 - 264130929754784*x^8 - 66381655299686400*x^7 - 66068412201811200*x^6 - 100204243327008000*x^5 + 33220715725209600*x^4 + 30566406996000000*x^3 + 3073502390976000*x^2 - 563930436960000*x - 70247748960000)[9] \\ Charles R Greathouse IV, Apr 07 2026

Crossrefs

Cf. A195284.

Sequence in context: A158781 A293299 A293486 * A063085 A020796 A011253

Adjacent sequences: A195395 A195396 A195397 * A195399 A195400 A195401

Keyword

nonn,cons

Author

Clark Kimberling, Sep 17 2011

Extensions

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

Status

approved