login
A374623
Decimal expansion of the diameter of the Spiral of Theodorus.
0
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.
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
KEYWORD
nonn,cons
AUTHOR
Gonzalo Martínez, Jul 15 2024
STATUS
approved