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A246966 Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice. 0
1, 5, 4, 2, 2, 1, 9, 7, 2, 1, 7, 0, 6, 5, 0, 5, 2, 5, 8, 5, 3, 1, 4, 1, 5, 7, 6, 4, 3, 6, 4, 2, 4, 5, 2, 9, 5, 6, 1, 9, 4, 8, 0, 7, 3, 5, 9, 1, 3, 1, 5, 4, 7, 8, 5, 3, 8, 8, 1, 6, 4, 0, 1, 9, 0, 8, 6, 3, 2, 1, 8, 1, 9, 3, 6, 7, 6, 9, 6, 7, 4, 8, 2, 3, 3, 9, 1, 1, 3, 1, 8, 7, 4, 4, 3, 6, 8, 0, 7, 5, 0, 2, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The ionic hexagonal (triangular) lattice considered here consists of three interpenetrating hexagonal lattices of ions with charges +1, -1, 0. Equivalently, one may consider the honeycomb net consisting of two hexagonal lattices of ions with charges +1 and -1 (the h-BN layer structure). In any case, this lattice sum is based on the nearest neighbor distance (not the length of the period of the ionic crystal structure, which is sqrt(3) times greater). - Andrey Zabolotskiy, Jun 21 2022
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 78.
LINKS
David Borwein, Jonathan M. Borwein and Keith F. Taylor, Convergence of lattice sums and Madelung's constant, J. Math. Phys. 26 (1985), 2999-3009.
Eric Weisstein's MathWorld, Madelung Constants
FORMULA
H_2 = (-3 + sqrt(3))*zeta(1/2)*((1 - sqrt(2))*zeta(1/2, 1/3) + zeta(1/2, 1/6)), where zeta(s,a) gives the generalized Riemann zeta function.
EXAMPLE
1.54221972170650525853141576436424529561948...
MATHEMATICA
H2 = (-3 + Sqrt[3])*Zeta[1/2]*((1 - Sqrt[2])*Zeta[1/2, 1/3] + Zeta[1/2, 1/6]); RealDigits[H2, 10, 103] // First
PROG
(PARI) (sqrt(3)-3)*zeta(1/2)*((1-sqrt(2))*zetahurwitz(1/2, 1/3) + zetahurwitz(1/2, 1/6)) \\ Charles R Greathouse IV, Jan 31 2018
CROSSREFS
Sequence in context: A242600 A102593 A090462 * A081749 A074825 A225063
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)