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A290705 Theta series of triamond.

%I A290705 #21 Sep 23 2021 05:18:33
%S A290705 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,
%T A290705 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,
%U A290705 36,0,24,0,18,0,30,0,36,12,12,0,42,0,24,48,12,30
%N A290705 Theta series of triamond.
%C A290705 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.
%C A290705 A004013 and 3*A045828, interleaved.
%H A290705 Toshikazu Sunada, <a href="https://www.ams.org/notices/200802/tx080200208p.pdf">Crystals that nature might miss creating</a>, Notices Amer. Math. Soc. 55 (No. 2, 2008), 208-215.
%H A290705 Toshikazu Sunada, <a href="/A038620/a038620_1.png">Correction to "Crystals That Nature Might Miss Creating"</a>, Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343. [Annotated scanned copy]
%H A290705 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laves_graph">Laves graph</a>
%t A290705 (* count lattice sites straightforwardly *)
%t A290705 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 *)
%t A290705 n = 10;
%t A290705 s = O[q]^(n^2 + 1) + Sum[q^(8 Norm[a + {i, j, k}]^2), {i, -n-1, n+1}, {j, -n-1,  n+1}, {k, -n-1, n+1}, {a, cell}];
%t A290705 CoefficientList[Normal[s], q] &
%t A290705 (* or use the generation function *)
%t A290705 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}];
%Y A290705 Cf. A004013, A005925, A045828, A113062.
%Y A290705 See A038620 for coordination sequence.
%K A290705 nonn
%O A290705 0,2
%A A290705 _Andrey Zabolotskiy_, Aug 09 2017