OFFSET
0,4
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
REFERENCES
C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Shigeo Naya, On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices, Progress of Theoretical Physics, 11 (1954), 53-62.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
FORMULA
G.f.: (1 - 16 * x^3 / ((1+3*x) * (1-x)^3))^(1/8) [Shigeo Naya]. - Andrey Zabolotskiy, Jun 01 2022
a(n) ~ (-1)^n * 3^n / (Gamma(1/8) * 2^(1/4) * n^(7/8)) * (1 - (-1)^n * sqrt(sqrt(2) - 1) * Gamma(1/8)^2 / (2^(13/4) * Pi * n^(1/4))). - Vaclav Kotesovec, Apr 27 2024
MATHEMATICA
CoefficientList[Series[(1 - 16 * x^3 / ((1+3*x) * (1-x)^3))^(1/8), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 27 2024 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
Offset changed, signs of terms changed, and more terms added by Andrey Zabolotskiy, Jun 01 2022
STATUS
approved