

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
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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 hBN 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



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)*((1sqrt(2))*zetahurwitz(1/2, 1/3) + zetahurwitz(1/2, 1/6)) \\ Charles R Greathouse IV, Jan 31 2018


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



