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A195403
Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
5
6, 9, 2, 0, 2, 8, 6, 7, 8, 4, 7, 1, 6, 5, 1, 7, 6, 7, 9, 0, 4, 3, 2, 8, 7, 4, 5, 2, 5, 6, 2, 9, 3, 2, 5, 2, 0, 0, 9, 4, 0, 2, 2, 7, 5, 9, 3, 1, 3, 3, 3, 2, 2, 7, 0, 3, 7, 6, 1, 6, 4, 8, 0, 3, 3, 1, 9, 2, 5, 7, 7, 4, 5, 6, 5, 6, 6, 8, 8, 7, 5, 7, 5, 3, 6, 4, 5, 9, 7, 8, 4, 0, 1, 8, 6, 1, 7, 5, 7, 8, 9
OFFSET
0,1
COMMENTS
See A195284 for definitions and a general discussion.
EXAMPLE
(A)=0.69202867847165176790432874525629325200...
MATHEMATICA
a = 1; b = Sqrt[c]; c = (1 + Sqrt[5])/2;
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) A195403 *)
N[x2, 100]
RealDigits[%] (* (B) A195404 *)
N[x3, 100]
RealDigits[%] (* (C) A195405 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* Philo(ABC, I) A195406 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 17 2011
EXTENSIONS
a(99) corrected by Georg Fischer, Jul 18 2021
STATUS
approved