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 A300072 #34 Jul 26 2025 18:42:23 %S A300072 9,4,5,0,2,6,8,1,9,1,3,1,9,8,1,9,0,6,2,2,8,5,0,4,6,4,8,0,5,1,5,6,4,8, %T A300072 0,4,7,1,7,9,5,8,6,1,0,8,2,2,9,2,9,5,5,5,3,7,6,0,4,4,5,0,2,6,2,2,2,7, %U A300072 9,0,1,9,1,7,7,4,8,5,2,3,0,7,6,8,7,9,5,7,0,9,5,8,8,9,2,5,6,9,8 %N A300072 Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations. %C A300072 The system of three equations is %C A300072 x + y + z = 10, %C A300072 x*z = y^2, %C A300072 x^2 + y^2 = z^2. %C A300072 See A300070 for the Havil reference and links to Abū Kāmil, who considered this system. This real solution was not given in Havil's book. %C A300072 This solution is x = x2 := 10*A248750, -y = -y2 = present entry, z = z2 = A300073. %C A300072 The other real solution with positive y is x = 10*A248752, y = A300070, z = A300071. %C A300072 Note that X2 = x2/5, -Y2 = -y2/5 and Z2 = z2/5 solve the system of equations (i) X2 + Y2 + Z2 = 2, (ii) X2*Z2 = (Y2)^2 and (iii) (X2)^2 + (Y2)^2 = (Z2)^2. %F A300072 -y2 = 5*(1 - phi - sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622. %F A300072 The minimal polynomial is x^4 + 10*x^3 - 50*x^2 - 1000*x - 2500. - _Joerg Arndt_, Jul 21 2025 %e A300072 -y2 = 9.450268191319819062285046480515648047179586108229295553760445026222... %e A300072 -y2/5 = 1.8900536382639638124570092961031296094359172216458591107520890052... %t A300072 RealDigits[5 (1 - GoldenRatio - Sqrt[GoldenRatio]), 10, 100][[1]] (* _Bruno Berselli_, Mar 02 2018 *) %Y A300072 Cf. A001622, A248750, A248752, A300070, A300071, A300073. %K A300072 nonn,cons,easy %O A300072 1,1 %A A300072 _Wolfdieter Lang_, Mar 02 2018