cp's OEIS Frontend

A321215 Decimal expansion of C[11] coefficient (negated) in 1/N expansion of lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.

%I A321215 #48 Jan 20 2019 09:56:37
%S A321215 6,0,1,6,3,3,5,7,1,7,6,9,0,3,4,6,8,2,9,2,2,1,8,5,3,3,1,5,0,7,5,4,5,4,
%T A321215 8,1,1,5,3,0,9,7,2,1,8,0,6,1,7,3,1,0,1,7,7,9,9,3,3,1,4,4,7,6,1,0,4,5,
%U A321215 4,6,1,0,0,8,9,6,7,6,1,2,6,1,7,3,9,5,2,4,3,2,9,2,1,2,9,2,5,4,0,9,0,8,4,7,4,5
%N A321215 Decimal expansion of C[11] coefficient (negated) in 1/N expansion of lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.
%C A321215 This is the 11th coefficient C[11] = -6016.337... in the 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of the N-sided, Pi-area regular polygon.
%C A321215 In context, the eigenvalue expression for the N-sided, Pi-area regular polygon is L = L0*(1 + C[3]/N^3 + C[5]/N^5 + C[6]/N^6 + C[7]/N^7 + C[8]/N^8 + ... + C[11]/N^11 + ...) where L0 = [A115368]^2 = [A244355] is the eigenvalue in the Pi-area circle.
%C A321215 C[11] was computed by first computing several hundred 200-digit eigenvalues in the range from N = 1000 to 3000, and then using linear regression to determine this expansion coefficient. All digits reported are correct. This is the first coefficient that appears to break the regular pattern involving roots of Bessel functions and Riemann zeta functions, for example, C[3] = 4*zeta(3) and C[5] = (12-2*L0)*zeta(5), where zeta(n) is the Riemann zeta function. C[11] is negative.
%H A321215 Robert Stephen Jones, <a href="/A321215/b321215.txt">Table of n, a(n) for n = 4..132</a> (sign corrected by _Georg Fischer_, Jan 20 2019)
%H A321215 Mark Boady, <a href="http://hdl.handle.net/1860/idea:6852">Applications of Symbolic Computation to the Calculus of Moving Surfaces</a>. PhD thesis, Drexel University, Philadelphia, PA. 2015.
%H A321215 P. Grinfeld and G. Strang, <a href="https://doi.org/10.1016/j.jmaa.2011.06.035">Laplace eigenvalues on regular polygons: A series in 1/N</a>, J. Math. Anal. Appl., 385-149, 2012.
%H A321215 Robert Stephen Jones, <a href="https://doi.org/10.1007/s10444-017-9527-y">Computing ultra-precise eigenvalues of the Laplacian within polygons</a>. Advances in Computational Mathematics, May 2017.
%H A321215 Robert Stephen Jones, <a href="https://arxiv.org/abs/1712.06082">The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions</a>, arXiv:1712.06082 [math.NA], 2017.
%e A321215 6016.335717690346829221853315075454811530972180617310177993314476104546100896...
%Y A321215 Cf. A321216 = C[12], the next coefficient in the 1/N expansion.
%Y A321215 Cf. A115368, A244355, A002117, and A013663.
%K A321215 nonn,cons
%O A321215 4,1
%A A321215 _Robert Stephen Jones_, Oct 31 2018