This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A192918 #57 Oct 06 2024 08:32:57 %S A192918 5,4,3,6,8,9,0,1,2,6,9,2,0,7,6,3,6,1,5,7,0,8,5,5,9,7,1,8,0,1,7,4,7,9, %T A192918 8,6,5,2,5,2,0,3,2,9,7,6,5,0,9,8,3,9,3,5,2,4,0,8,0,4,0,3,7,8,3,1,1,6, %U A192918 8,6,7,3,9,2,7,9,7,3,8,6,6,4,8,5,1,5,7,9,1,4,5,7,6,0,5,9,1,2,5,4,6,2 %N A192918 Decimal expansion of the real root of r^3 + r^2 + r - 1. %C A192918 The real solution r of the cubic equation r^3 + r^2 + r - 1 = 0 is the reciprocal of the tribonacci constant A058265. If the four sides of a quadrilateral form a geometric progression 1:r:r^2:r^3 where r is the common ratio then r is limited to the range 1/t < r < t where t is the tribonacci constant. More generally if f(n) is the n-th step Fibonacci constant then a polygon of n+1 sides can have sides in a geometric progression 1:r:r^2:...:r^n if the common ratio r is limited to the range 1/f(n) < r < f(n). %C A192918 From _Wolfdieter Lang_, Aug 22 2022: (Start) %C A192918 The roots of this cubic are obtained from the roots of y^3 + (2/3)*y - 34/27 after subtracting 1/3. The y-roots are y1 = (u_p^(1/3) + u_m^(1/3)*e_m)/3, y2 = (e_m*u_p^(1/3) + u_m^(1/3))/3 and y3 = e_p*(u_p^(1/3) + u_m^(1/3))/3. Here u_p = 17 + 3*sqrt(33), u_m = 17 - 3*sqrt(33), e_p = -(1 + sqrt(3)*i) and e_m = -(1 - sqrt(3)*i), where i = sqrt(-1). %C A192918 The roots of the x-cubic are then x1, the present real solution, and x2 = y2 - 1/3 = -0.771844506... + 1.11514250...*i and the complex conjugate x3 = y3 - 1/3. (End) %H A192918 G. C. Greubel, <a href="/A192918/b192918.txt">Table of n, a(n) for n = 0..10000</a> %H A192918 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 38.9, A function encoding the Hilbert curve, page 748, y_1. %H A192918 Wikipedia, <a href="http://en.wikipedia.org/wiki/Golden_ratio#Triangle_whose_sides_form_a_geometric_progression">Triangle whose sides form a geometric progression</a>. %H A192918 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a> %F A192918 Equals (1/3)*(-1-2/(17+3*sqrt(33))^(1/3) + (17+3*sqrt(33))^(1/3)). %F A192918 Equals (1/3)*(u_p^(1/3) + u_m^(1/3)*e_m - 1), with u_p = 17 + 3*sqrt(33), u_m = 17 - 3*sqrt(33), and e_m = -(1 - sqrt(3)*i), with i = sqrt(-1). - _Wolfdieter Lang_, Aug 22 2022 %F A192918 Equals hypergeom([1/4,1/2,3/4],[2/3,4/3],16/27)/2. - _Gerry Martens_, Jul 13 2023 %e A192918 0.543689012692076361570855971801747986525203297650983935240... %t A192918 N[Reduce[r+r^2+r^3==1, r], 100] %t A192918 RealDigits[(1/3)*(-1 -2/(17+3*Sqrt[33])^(1/3) +(17+3*Sqrt[33])^(1/3)), 10, 100][[1]] (* _G. C. Greubel_, Feb 06 2019 *) %t A192918 RealDigits[Root[r^3+r^2+r-1,1],10,120][[1]] (* _Harvey P. Dale_, May 18 2023 *) %o A192918 (PARI) polrootsreal(r^3 + r^2 + r - 1)[1] \\ _Charles R Greathouse IV_, Apr 14 2014 %o A192918 (Magma) SetDefaultRealField(RealField(100)); (1/3)*(-1 -2/(17 +3*Sqrt(33))^(1/3) +(17+3*Sqrt(33))^(1/3)); // _G. C. Greubel_, Feb 06 2019 %o A192918 (Sage) numerical_approx((1/3)*(-1 -2/(17+3*sqrt(33))^(1/3) +(17+ 3*sqrt(33))^(1/3)), digits=100) # _G. C. Greubel_, Feb 06 2019 %Y A192918 Reciprocal of A058265. %Y A192918 Cf. A376841. %K A192918 nonn,cons %O A192918 0,1 %A A192918 _Frank M Jackson_, Aug 26 2011