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A321215
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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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2
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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
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OFFSET
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4,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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6016.335717690346829221853315075454811530972180617310177993314476104546100896...
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CROSSREFS
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Cf. A321216 = C[12], the next coefficient in the 1/N expansion.
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KEYWORD
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AUTHOR
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STATUS
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approved
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