A373785 Decimal expansion of the area of the Spiral of Theodorus.

2, 2, 2, 3, 4, 5, 9, 8, 3, 0, 0, 0, 7, 1, 3, 0, 1, 0, 7, 8, 5, 4, 5, 7, 7, 0, 1, 8, 7, 4, 4, 7, 9, 9, 4, 7, 3, 3, 0, 7, 0, 5, 6, 9, 0, 9, 1, 3, 7, 1, 3, 8, 3, 2, 7, 3, 1, 4, 3, 6, 7, 1, 8, 5, 6, 3, 8, 0, 1, 5, 5, 4, 4, 5, 1, 1, 1, 6, 6, 7, 6, 7, 8, 1, 3, 1, 5, 3, 7, 8, 7, 6, 3, 9, 5, 8, 8, 5, 6, 5, 9, 4, 1

Offset: 1

Author

Gonzalo Martínez, Jun 23 2024

Keywords

• nonn
• cons

Comments

The length of the legs of the k-th triangle of the Spiral of Theodorus are 1 and sqrt(k), while the length of the hypotenuse is sqrt(k + 1). Theodorus stopped at the 16th triangle, whose hypotenuse measures sqrt(17), because from the next triangle his spiral began to overlap (see A072895).

The area of the k-th triangle is (1/2) * sqrt(k), so the area A of the spiral constructed by Theodorus is A = (1/2)*Sum_{k=1..16} sqrt(k) = 22.234598300071...

The perimeter of the spiral of Theodorus (up to the 16th triangle) is 17 + sqrt(17).

Examples

			22.234598300071301078545770187447994733070569...
		

Links

• Wikipedia, Spiral of Theodorus.

Crossrefs

Cf. A072895.

Programs

Mathematica

RealDigits[(1/2)*Sum[Sqrt[k],{k,16}],10,120][[1]] (* Stefano Spezia, Jun 23 2024 *)

Formula

Equals (1/2)*Sum_{k=1..16} sqrt(k).