A072895 -id:A072895 - cp's OEIS frontend

A351861 Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.

2, 1, -1, -1, 5, 1, -521, -29, 1067, 13221, -538019, -692393, 2088537, 3155999, -27611845, -33200670659, 1202005038007, 40366435189, -29289910899229, -14754517273097, 1825124640773023, 18449097055233961, -250479143430425927, -1976767636081931863, 1419438523008706978221

Offset: 0

Robert B Fowler, Feb 22 2022

S(i) is the sum of the angles of the first i-1 triangles in the Spiral of Theodorus (in radians). [Corrected by Robert B Fowler, Oct 23 2022]
S(i) = K + sqrt(i) * (2 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...) where K is Hlawka's Schneckenkonstante, K = A105459 * (-1) = -2.1577829966... .
The coefficients in the polynomial series are a(n)/A351862(n). The series is asymptotic, but is very accurate even for low values of i.

			2/1 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

David Brink, The Spiral of Theodorus and sums of zeta values at half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pages 779-786.
David Brink, The Spiral of Theodorus and sums of zeta values at half-integers, July 2012. There are errors in equation (9): the denominator factors n, n^2, n^3, n^4, n^5 should actually be n, n^3, n^5, n^7, n^9, respectively.
Detlef Gronau, The Spiral of Theodorus, The American Mathematical Monthly, Vol. 111, No. 3 (March 2004), pages 230-237.
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math. 89 (1980) pages 19-44. [For a summary in English, see the Davis reference, pages 157-167.]
Herbert Kociemba, The Spiral of Theodorus, 2018.
Wikipedia, Spiral of Theodorus

Cf. A351862 (denominators).
Cf. A105459, A185051 (Hlawka's constant).
Cf. A027641, A027642 (Bernoulli numbers).
Cf. A072895, A224269 (spiral revolutions).

c[0] = 2; c[n_] := ((2*n - 2)!/(n - 1)!) * Sum[(-1)^(n + 1) * BernoulliB[n - k] * k!/(4^(n - k - 1) * (2*k + 1)! * (n - k)!), {k, 0, n}]; Numerator @ Array[c, 30, 0] (* Amiram Eldar, Feb 22 2022 *)

a(n) = {numerator(if(n==0, 2, ((2*n-2)!/(n-1)!) * sum(k=0, n, (-1)^(n+1) * bernfrac(n-k) * k! / (4^(n-k-1) * (2*k+1)! * (n-k)!))))} \\ Andrew Howroyd, Feb 22 2022

Let r(n) = ((2*n-2)! / (n-1)!) * Sum_{k=0..n} ((-1)^(n+1)*B(n-k)*k!) / ((4^(n-k-1) * (2*k+1)! * (n-k)!) ) for n > 0, where B(n-k) are Bernoulli numbers. Then:

a(n) = numerator(r(n)) for n >= 1 and additionally a(0) = 2.

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

Original entry on oeis.org

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

Gonzalo Martínez, Jun 23 2024

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).

			22.234598300071301078545770187447994733070569...

Wikipedia, Spiral of Theodorus.

Cf. A072895.

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

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

cp's OEIS Frontend

A351861 Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula