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%I #37 Jun 21 2022 14:58:04
%S 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,
%T 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,
%U 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
%N Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice.
%C 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
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 78.
%H David Borwein, Jonathan M. Borwein and Keith F. Taylor, <a href="https://doi.org/10.1063/1.526675">Convergence of lattice sums and Madelung's constant</a>, J. Math. Phys. 26 (1985), 2999-3009.
%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>
%F 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.
%e 1.54221972170650525853141576436424529561948...
%t 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
%o (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
%Y Cf. A088537, A085469, A090734, A247040.
%K nonn,cons,easy
%O 1,2
%A _Jean-François Alcover_, Sep 10 2014
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