A245263 - OEIS

%I #24 Oct 27 2023 10:36:36

%S 3,8,9,1,1,5,6,8,2,3,3,2,6,8,5,3,8,1,8,0,7,8,2,6,2,5,5,6,7,1,9,9,0,5,

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

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

%N Decimal expansion of the minimum total length of lines tying together the five points of a regular pentagon having unit sides.

%H Jean-François Alcover, <a href="/A245263/a245263.gif">Pentagon Steiner tree.</a> After James C. Bergquist.

%H Herman P. Robinson, <a href="http://www.escholarship.org/uc/item/2t95c0bp">Mathematical Constants.</a> p. 182.

%H <a href="/index/Al#algebraic_08">Index entries for algebraic numbers, degree 8</a>

%F (7*sqrt(3)-sqrt(15))/12 + sqrt(18-6*sqrt(5))/3 + (1+sqrt(5))*sqrt(10-2*sqrt(5))/8 + sqrt(5+2*sqrt(5))/2 = sqrt(17+7*sqrt(5)+sqrt(6*(65+29*sqrt(5))))/2.

%e 3.891156823326853818078262556719905049852981445670139299627728956...

%t (7*Sqrt[3]-Sqrt[15])/12 + Sqrt[18-6*Sqrt[5]]/3 + (1+Sqrt[5])*Sqrt[10-2*Sqrt[5]]/8 + Sqrt[5+2*Sqrt[5]]/2 // RealDigits[#, 10, 101] & // First

%o (PARI) sqrt(polrootsreal(x^4-17*x^3+29*x^2-13*x+1)[4]) \\ _Charles R Greathouse IV_, Jul 15 2014

%o (PARI) polrootsreal(x^8-17*x^6+29*x^4-13*x^2+1)[8] \\ _Charles R Greathouse IV_, Oct 27 2023

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jul 15 2014