cp's OEIS Frontend

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

%I A351861 #46 Aug 11 2025 08:30:05
%S A351861 2,1,-1,-1,5,1,-521,-29,1067,13221,-538019,-692393,2088537,3155999,
%T A351861 -27611845,-33200670659,1202005038007,40366435189,-29289910899229,
%U A351861 -14754517273097,1825124640773023,18449097055233961,-250479143430425927,-1976767636081931863,1419438523008706978221
%N A351861 Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.
%C A351861 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]
%C A351861 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... .
%C A351861 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.
%D A351861 P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.
%H A351861 David Brink, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.119.09.779">The Spiral of Theodorus and sums of zeta values at half-integers</a>, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pages 779-786.
%H A351861 David Brink, <a href="http://davidbrink.dk/Theodoros.pdf">The Spiral of Theodorus and sums of zeta values at half-integers</a>, 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.
%H A351861 Detlef Gronau, <a href="https://www.jstor.org/stable/4145130">The Spiral of Theodorus</a>, The American Mathematical Monthly, Vol. 111, No. 3 (March 2004), pages 230-237.
%H A351861 Edmund Hlawka, <a href="https://gdz.sub.uni-goettingen.de/id/PPN362162050_0089">Gleichverteilung und Quadratwurzelschnecke</a>, Monatsh. Math. 89 (1980) pages 19-44. [For a summary in English, see the Davis reference, pages 157-167.]
%H A351861 Herbert Kociemba, <a href="http://kociemba.org/themen/spirale/theodorus.html">The Spiral of Theodorus</a>, 2018.
%H A351861 Wikipedia, <a href="https://en.wikipedia.org/wiki/Spiral_of_Theodorus">Spiral of Theodorus</a>
%F A351861 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:
%F A351861 a(n) = numerator(r(n)) for n >= 1 and additionally a(0) = 2.
%e A351861 2/1 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...
%t A351861 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 *)
%o A351861 (PARI) 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
%Y A351861 Cf. A351862 (denominators).
%Y A351861 Cf. A105459, A185051 (Hlawka's constant).
%Y A351861 Cf. A027641, A027642 (Bernoulli numbers).
%Y A351861 Cf. A072895, A224269 (spiral revolutions).
%K A351861 sign,frac,easy
%O A351861 0,1
%A A351861 _Robert B Fowler_, Feb 22 2022