A228475 Positive real root of 37*x^4+36*x^3+6*x^2-12*x-3.

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

Offset

0,1

Comments

A Soddyian triangle is a triangle whose outer Soddy circle has degenerated into a straight line. Its side lengths are related by the equation 1/sqrt(s-c)=1/sqrt(s-b)+1/sqrt(s-a) where the sides a<=b<=c and s is the semiperimeter. If the side lengths of such a triangle form an arithmetic progression 1, 1+d, 1+2d, where d is the common difference, then d = 0.5165877... and is the solution to the equation 37d^4+36d^3+6d^2-12d-3 = 0 such that 0<d<1. This triangle has angles of approx. 105.96, 45.82 and 28.22 degs.

Links

Table of n, a(n) for n=0..82.

F. M. Jackson, Soddyian triangles, Forum Geometr. 13 (2013), 1-6.

Index entries for algebraic numbers, degree 4

Formula

d = (-18+16*sqrt(3)+37*sqrt((608*sqrt(3))/1369-240/1369))/74.

Example

0.51658772215405264712532988077485052478638588883477756993492758314966...

Mathematica

a=1; b=1+d; c=1+2d; s=(a+b+c)/2; sol=Solve[1/Sqrt[s-a]+1/Sqrt[s-b]-1/Sqrt[s-c]==0&&0<d<1, d]; RealDigits[N[d /. sol[[1]], 100]][[1]]

Prog

(PARI) polrootsreal(37*x^4+36*x^3+6*x^2-12*x-3)[2] \\ Charles R Greathouse IV, Apr 16 2014

Crossrefs

Cf. A210484.

Sequence in context: A318553 A176909 A131944 * A296355 A306700 A058651

Adjacent sequences: A228472 A228473 A228474 * A228476 A228477 A228478

Keyword

nonn,cons

Author

Frank M Jackson, Aug 23 2013

Status

approved