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A351861
Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.
1
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
0,1
COMMENTS
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.
REFERENCES
P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.
LINKS
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.
FORMULA
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.
EXAMPLE
2/1 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A351862 (denominators).
Cf. A105459, A185051 (Hlawka's constant).
Cf. A027641, A027642 (Bernoulli numbers).
Cf. A072895, A224269 (spiral revolutions).
Sequence in context: A327778 A099940 A157249 * A343233 A155586 A069739
KEYWORD
sign,frac,easy
AUTHOR
Robert B Fowler, Feb 22 2022
STATUS
approved