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A321216
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Decimal expansion of C[12] coefficient in 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.
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2
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2, 5, 2, 0, 0, 9, 7, 3, 7, 9, 2, 9, 3, 2, 4, 6, 4, 6, 7, 6, 0, 6, 5, 2, 1, 2, 2, 3, 9, 5, 3, 8, 5, 4, 7, 7, 0, 2, 8, 7, 8, 0, 6, 5, 3, 2, 2, 5, 5, 6, 6, 1, 4, 6, 4, 9, 7, 9, 0, 1, 5, 3, 9, 4, 4, 7, 7, 3, 6, 0, 5, 4, 2, 4, 0, 2, 9, 8, 2, 8, 3, 6, 7, 4, 5, 6, 6, 2, 0, 7, 3, 7, 1, 3, 4, 1, 5, 7, 8, 5
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OFFSET
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5,1
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COMMENTS
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This is the 12th coefficient C[12] in the 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of Pi-area, N-sided regular polygon. It was determined using experimental mathematics by computing the coefficient to 125 digits of precision. It can be computed using the expression in the Formula section. It is expressed in terms of L0 = [A115368]^2 = [A244355] = 5.78318... (eigenvalue of unit-radius circle) and Riemann zeta functions. Although this is derived using experimental mathematics, the decimal expansion reported is equal to that expression. 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[12]/N^12 + ...). The expression for this coefficient follows a pattern similar to lower-order coefficients (except C[11] [A321215]), e.g., C[3]=4*zeta(3) and C[5]=(12-2*L0)*zeta(5).
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LINKS
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EXAMPLE
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25200.9737929324646760652122395385477028780653225566146497901539447736054240...
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PROG
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(PARI) {default(realprecision, 100); L0=solve(x=2, 3, besselj(0, x))^2; (32/3+272*L0/3-16*L0^2)*zeta(3)^4+(1360/3-488*L0/3+456*L0^2+107*L0^3/3+5*L0^4/8)*zeta(3)*zeta(9)+(432-216*L0-207*L0^2+47*L0^3/2+11*L0^4/8)*zeta(5)*zeta(7)}
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CROSSREFS
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Cf. A321215 is decimal expansion of C[11], the next lower order coefficient.
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KEYWORD
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AUTHOR
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STATUS
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approved
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