A262823 -id:A262823 - cp's OEIS frontend

A263202 Decimal expansion of the lowest Dirichlet eigenvalue of the Laplacian within the unit-edged regular hexagon.

7, 1, 5, 5, 3, 3, 9, 1, 3, 3, 9, 2, 6, 0, 5, 5, 1, 2, 8, 2, 1, 0, 0, 1, 7, 6, 1, 6, 8, 3, 3, 1, 3, 9, 2, 8, 0, 6, 6, 9, 1, 9, 9, 5, 8, 5, 7, 7, 6, 9, 7, 7, 9, 2, 0, 3, 4, 9, 4, 2, 4, 9, 0, 4, 7, 4, 4, 3, 3, 3, 1, 2, 2, 5, 0, 9, 2, 5, 3, 3, 7, 5, 4, 8, 7, 5

Offset: 1

Robert Stephen Jones, Oct 12 2015

			7.1553391339260551282100176168331392806691995857769779...

Robert Stephen Jones, Table of n, a(n) for n = 1..1001
L. Bauer and E. L. Reiss, Cutoff wavenumbers and modes of hexagonal waveguides, SIAM J. of Appl. Math., 35 (1978), 508-514. (Note: 6-digit results.)
L. M. Cureton and J. R. Kuttler, Eigenvalues of the Laplacian on regular polygons and polygons resulting from their dissection, Journal of Sound and Vibration, 220 (1998), 83-98. (Note: Table 2 presents their 8-digit digit results.)
Robert S. Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons, arXiv preprint arXiv:1602.08636, 2016

Cf. A262701 (L-shape) and A262823 (regular pentagon).

A276813 Continued fraction for the ratio of the lowest two Dirichlet eigenvalues of the Laplacian within the regular pentagon.

Original entry on oeis.org

2, 1, 1, 8, 1, 4, 2, 2, 16, 3, 3, 1, 3, 1, 1, 2, 1, 1, 3, 1, 3, 3, 5, 1, 13, 1, 4, 1, 1, 13, 4, 3, 1, 4, 1, 1, 6, 5, 9, 1, 13, 2, 15, 1, 2, 3, 3, 1, 4, 9, 2, 14, 1, 4, 1, 7, 1, 1, 11, 1, 4, 5, 2, 3, 2, 1, 14, 1, 1, 2, 1, 1, 1, 1, 20, 3, 2, 1, 2, 2, 7, 1, 2

Offset: 1

Robert Stephen Jones, Sep 18 2016

The eigenvalues of the Laplacian within the regular pentagon with Dirichlet boundary conditions are calculated to at least 1000 digits. The ratio of the second eigenvalue to the first is calculated and expressed as a continued fraction. The ratio has an advantage since it is independent of the pentagon area. All terms in this expansion are correct.

			2.52683872... = 2+1/(1+1/(1+1/(8+1/(1+...)))).

Robert Stephen Jones, Table of n, a(n) for n = 1..978

Cf. A262823.

cp's OEIS Frontend

A263202 Decimal expansion of the lowest Dirichlet eigenvalue of the Laplacian within the unit-edged regular hexagon.

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