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

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, 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, 6, 6, 5, 4, 0, 5, 6, 0, 1, 8, 2

Offset

1,1

Comments

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.

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)).

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...

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.

Links

Table of n, a(n) for n=1..79.

Wikipedia, Spiral of Theodorus.

Formula

Equals sqrt(24 - 2*sqrt(119)*cos(Sum_{k=7..16} arctan(1/sqrt(k)))).

Example

6.733656013142546631704124493808698228682375551979922...

Mathematica

RealDigits[Sqrt[24 - 2*Sqrt[119]*Cos[Sum[ArcTan[1/Sqrt[k]], {k, 7, 16}]]], 10, 120][[1]] (* Amiram Eldar, Aug 20 2024 *)

Crossrefs

Cf. A373785.

Sequence in context: A277135 A153628 A154972 * A093603 A245513 A105739

Adjacent sequences: A374620 A374621 A374622 * A374624 A374625 A374626

Keyword

nonn,cons

Author

Gonzalo Martínez, Jul 15 2024

Status

approved