A351862 Denominators of the coefficients in a series for the angles in the Spiral of Theodorus.

1, 6, 120, 840, 8064, 4224, 2196480, 199680, 5013504, 74088448, 1568931840, 1899233280, 2411724800, 2831155200, 8757706752, 6968215339008, 76890652016640, 1488206168064, 289223097712640, 74371653697536, 2197648866017280, 10176804748787712, 29785769996451840

Offset

0,2

Comments

S(i) is the sum of the angles in the first i-1 triangles of 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 = A105459 * (-1) = -2.1577829966... .

The coefficients in the polynomial series are A351861(n)/a(n). The series is asymptotic, but is accurate for even very low values of i.

See A351861 for the numerators, as well as references, links, and crossrefs.

Links

Table of n, a(n) for n=0..22.

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}]; Denominator @ Array[c, 30, 0] (* Amiram Eldar, Feb 22 2022 *)

Crossrefs

Cf. A351861 (numerators).

Sequence in context: A076231 A076234 A066289 * A170917 A115678 A048604

Adjacent sequences: A351859 A351860 A351861 * A351863 A351864 A351865

Keyword

nonn,frac,easy

Author

Robert B Fowler, Feb 22 2022

Status

approved