cp's OEIS Frontend

A374623 Decimal expansion of the diameter of the Spiral of Theodorus.

%I A374623 #45 Aug 20 2024 01:53:21
%S A374623 6,7,3,3,6,5,6,0,1,3,1,4,2,5,4,6,6,3,1,7,0,4,1,2,4,4,9,3,8,0,8,6,9,8,
%T A374623 2,2,8,6,8,2,3,7,5,5,5,1,9,7,9,9,2,2,0,9,1,0,4,9,0,3,0,0,9,8,1,6,2,8,
%U A374623 6,6,5,4,0,5,6,0,1,8,2
%N A374623 Decimal expansion of the diameter of the Spiral of Theodorus.
%C A374623 Let O be the origin of the Spiral of Theodorus and P_1, P_2, P_3, ..., P_17 its vertices, where the k-th triangle has vertices O, P_k, and P_(k+1) for 1 <= k <= 16. The diameter of the spiral is the greatest distance between the vertices, corresponding to the segment joining vertices P_7 and P_17. That is, Sup{dist(P_i, P_j) | 1 <= i,j <= 17} = P_7P_17.
%C A374623 For simplicity, let us rename A = P_7 and B = P_17, so the diameter is equal to the longest side of triangle AOB, where OA = sqrt(7), OB = sqrt(17), and the angle AOB measures w = Sum_{k=7..16} arctan(1/sqrt(k)), since the angle at vertex O of the k-th triangle is arctan(1/sqrt(k)).
%C A374623 Then, by the law of cosines, it follows that the diameter d of the Theodorus spiral is equal to d = sqrt(24 - 2*sqrt(119)*cos(w)) = 6.7336560131425466317...
%C A374623 On the other hand, it is known that the perimeter p of the spiral (up to the 16th triangle) is p = 17 + sqrt(17) (see A373785). Thus, it is observed that p/d = 3.1369445639..., so |Pi - p/d| < 0.005.
%H A374623 Wikipedia, <a href="https://en.wikipedia.org/wiki/Spiral_of_Theodorus">Spiral of Theodorus</a>.
%F A374623 Equals sqrt(24 - 2*sqrt(119)*cos(Sum_{k=7..16} arctan(1/sqrt(k)))).
%e A374623 6.733656013142546631704124493808698228682375551979922...
%t A374623 RealDigits[Sqrt[24 - 2*Sqrt[119]*Cos[Sum[ArcTan[1/Sqrt[k]], {k, 7, 16}]]], 10, 120][[1]] (* _Amiram Eldar_, Aug 20 2024 *)
%Y A374623 Cf. A373785.
%K A374623 nonn,cons
%O A374623 1,1
%A A374623 _Gonzalo Martínez_, Jul 15 2024