

A290705


Theta series of triamond.


1



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.


LINKS

Table of n, a(n) for n=0..72.
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. A005925, A113062.
Sequence in context: A092731 A201567 A161829 * A115456 A007386 A007385
Adjacent sequences: A290702 A290703 A290704 * A290706 A290707 A290708


KEYWORD

nonn


AUTHOR

Andrey Zabolotskiy, Aug 09 2017


STATUS

approved



