

A290705


Theta series of triamond.


2



1, 3, 0, 6, 0, 6, 8, 12, 6, 9, 0, 6, 0, 18, 0, 12, 12, 12, 0, 18, 0, 12, 24, 12, 8, 21, 0, 24, 0, 6, 0, 24, 6, 24, 0, 12, 0, 30, 24, 12, 24, 12, 0, 30, 0, 30, 0, 24, 24, 27, 0, 12, 0, 18, 32, 36, 0, 24, 0, 18, 0, 30, 0, 36, 12, 12, 0, 42, 0, 24, 48, 12, 30
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OFFSET

0,2


COMMENTS

Theta series with respect to a node of a lattice known as triamond, Laves graph [embedded in space], K_4 lattice, (10,3)a, or srs net. This lattice possesses the "strong isotropic" property; the only other lattice that has this property in 3 dimensions is the diamond lattice. Unlike diamond, triamond is chiral.
A004013 and 3*A045828, interleaved.


REFERENCES

Sunada, Toshikazu. "Crystals that nature might miss creating." Notices Amer. Math. Soc. 55 (No. 2, 2008), 208215.
Toshikazu Sunada, Correction to "Crystals That Nature Might Miss Creating", Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343.


LINKS

Table of n, a(n) for n=0..72.
Toshikazu Sunada, Correction to "Crystals That Nature Might Miss Creating", Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343. [Annotated scanned copy]
Wikipedia, Laves graph


MATHEMATICA

(* count lattice sites straightforwardly *)
cell = Join @@ ({#, # + {1, 1, 1}/2} & /@ {{0, 0, 0}, {1/4, 0, 1/4}, {1/4, 1/4, 0}, {0, 1/4, 1/4}}); (* lattice sites in a conventional bcc unit cell *)
n = 10;
s = O[q]^(n^2 + 1) + Sum[q^(8 Norm[a + {i, j, k}]^2), {i, n1, n+1}, {j, n1, n+1}, {k, n1, n+1}, {a, cell}];
CoefficientList[Normal[s], q] &
(* or use the generation function *)
a[n_] := SeriesCoefficient[ EllipticTheta[3, 0, x^8]^3 + EllipticTheta[ 2, 0, x^8]^3 + 3/4 EllipticTheta[3, 0, x^2] EllipticTheta[2, 0, x^2]^2, {x, 0, n}];


CROSSREFS

Cf. A004013, A005925, A045828, A113062.
See A038620 for coordination sequence.
Sequence in context: A092731 A201567 A161829 * A115456 A007386 A007385
Adjacent sequences: A290702 A290703 A290704 * A290706 A290707 A290708


KEYWORD

nonn,changed


AUTHOR

Andrey Zabolotskiy, Aug 09 2017


STATUS

approved



