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%I #24 Oct 27 2023 10:39:08
%S 2,9,0,7,7,2,9,7,8,9,6,9,4,1,1,3,8,3,7,1,3,2,5,9,6,6,9,7,1,6,4,5,2,2,
%T 2,9,6,4,8,3,1,8,9,5,5,0,0,6,1,4,6,7,3,0,3,5,8,4,6,0,1,4,8,5,1,2,8,2,
%U 7,6,9,2,9,8,4,0,8,0,0,7,4,2,1,7,7,0,4,6,0,1,6,9,8,2,9,5,1,6,4
%N Decimal expansion of the real positive root of the equation: 4*d^4 + 12*d^3 + 8*d^2 - 1 = 0.
%C If the side lengths of a triangle form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) where d is the common difference between the denominators of the harmonic progression, then when d = 0.290772978969... it forms a unique right triangle. The angles (in degrees) are approximately 39.2195, 50.7805, 90.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F d as given by the real positive root of 4*d^4 + 12*d^3 + 8*d^2 - 1 = 0.
%e 0.29077297896941138371325966971645222964831895500614673035846014851282...
%t N[Reduce[-1+8d^2+12d^3+4d^4==0, d], 100]
%t RealDigits[x/.FindRoot[4x^4+12x^3+8x^2-1==0,{x,.2}, WorkingPrecision-> 120]][[1]] (* _Harvey P. Dale_, Feb 15 2015 *)
%o (PARI) polrootsreal(4*x^4+12*x^3+8*x^2-1)[2] \\ _Charles R Greathouse IV_, Apr 15 2014
%K easy,nonn,cons
%O 0,1
%A _Frank M Jackson_, Aug 02 2011
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