A185051 -id:A185051 - cp's OEIS frontend

A105459 Decimal expansion of Hlawka's Schneckenkonstante K = -2.157782... (negated).

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

Offset: 1

David Brink, Jun 13 2011

			-2.157782996659446220929142786829577723504139598607562455...

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

Robert G. Wilson v, Table of n, a(n) for n = 1..16384
David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]
Herbert Kociemba, The Spiral of Theodorus.

Cf. A185051 for continued fraction expansion.

Cf. A072895, A137515, A352741.

evalf(Sum((-1)^k*Zeta(k + 1/2)/(2*k+1), k=0..infinity), 120); # Vaclav Kotesovec, Mar 01 2016

RealDigits[ NSum[(-1)^k*Zeta[k + 1/2]/(2 k + 1), {k, 0, Infinity}, Method -> "AlternatingSigns", AccuracyGoal -> 2^6, PrecisionGoal -> 2^6, WorkingPrecision -> 2^7], 10, 2^7][[1]] (* Robert G. Wilson v, Jul 11 2013 *)

sumalt(k=0,(-1)^k*zeta(k+1/2)/(2*k+1)) \\ M. F. Hasler, Mar 31 2022

Sum_{x=1..n-1} arctan(1/sqrt(x)) = 2*sqrt(n) + K + o(1). [Corrected by M. F. Hasler, Mar 31 2022]

Equals Sum_{k>=0} (-1)^k*zeta(k+1/2)/(2*k+1). - Robert B Fowler, Oct 23 2022

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A105459 Decimal expansion of Hlawka's Schneckenkonstante K = -2.157782... (negated).

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