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A373785
Decimal expansion of the area of the Spiral of Theodorus.
1
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
1,1
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).
FORMULA
Equals (1/2)*Sum_{k=1..16} sqrt(k).
EXAMPLE
22.234598300071301078545770187447994733070569...
MATHEMATICA
RealDigits[(1/2)*Sum[Sqrt[k], {k, 16}], 10, 120][[1]] (* Stefano Spezia, Jun 23 2024 *)
CROSSREFS
Cf. A072895.
Sequence in context: A359319 A357384 A022865 * A089150 A056697 A132427
KEYWORD
nonn,cons
AUTHOR
Gonzalo Martínez, Jun 23 2024
STATUS
approved