cp's OEIS Frontend

A246966 Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice.

%I A246966 #39 Sep 18 2024 10:50:19
%S A246966 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 A246966 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 A246966 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 A246966 Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice.
%C A246966 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 A246966 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 78.
%H A246966 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 A246966 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>
%F A246966 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 A246966 1.54221972170650525853141576436424529561948...
%t A246966 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 A246966 (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 A246966 Cf. A088537, A085469, A090734, A247040.
%K A246966 nonn,cons,easy
%O A246966 1,2
%A A246966 _Jean-François Alcover_, Sep 10 2014