A321216 -id:A321216 - 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.

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, 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, 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

Offset: 4

Robert Stephen Jones, Oct 31 2018

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.
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[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.

			6016.335717690346829221853315075454811530972180617310177993314476104546100896...

Robert Stephen Jones, Table of n, a(n) for n = 4..132 (sign corrected by _Georg Fischer_, Jan 20 2019)
Mark Boady, Applications of Symbolic Computation to the Calculus of Moving Surfaces. PhD thesis, Drexel University, Philadelphia, PA. 2015.
P. Grinfeld and G. Strang, Laplace eigenvalues on regular polygons: A series in 1/N, J. Math. Anal. Appl., 385-149, 2012.
Robert Stephen Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons. Advances in Computational Mathematics, May 2017.
Robert Stephen Jones, The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions, arXiv:1712.06082 [math.NA], 2017.

Cf. A321216 = C[12], the next coefficient in the 1/N expansion.

Cf. A115368, A244355, A002117, and A013663.

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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.

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