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

Author

Robert B Fowler, Feb 22 2022

Keywords

• nonn
• frac
• easy

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.

Examples

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

Crossrefs

Cf. A351861 (numerators).

Programs

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 *)