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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 (list; graph; refs; listen; history; text; internal format)
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), 208-215.

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, -n-1, n+1}, {j, -n-1,  n+1}, {k, -n-1, 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

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Last modified October 20 06:51 EDT 2018. Contains 316378 sequences. (Running on oeis4.)