A005890 Theta series of hexagonal close-packing with respect to center of triangle between two layers. (Formerly M2195)

0, 0, 0, 3, 0, 0, 1, 0, 0, 3, 0, 1, 2, 0, 0, 4, 0, 2, 2, 2, 2, 2, 1, 2, 1, 1, 0, 4, 0, 0, 0, 2, 1, 6, 2, 4, 1, 2, 1, 2, 0, 5, 2, 3, 1, 6, 0, 4, 0, 4, 2, 2, 2, 4, 0, 2, 0, 5, 2, 2, 2, 4, 0, 2, 1, 4, 3, 5, 2, 2, 0, 2, 2, 9, 2, 6, 3, 6, 0, 4, 2, 2, 3, 8, 2, 2, 1

Offset

0,4

Comments

The triangle separates a tetrahedron and an octahedron.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534. See page 6530 equation 66.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of x^3 * ( f(x^3, x^15) * (f(x^16, x^32) * f(x^15, x^39) + x^6 * f(x^8, x^40) * f(x^3, x^51)) + f(x^6, x^12) * (f(x^16, x^32) * f(x^12, x^42) + f(x^8, x^40) * f(x^24, x^30)) ) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 11 2018

G.f.: Sum{i, j, k in Z} x^(9*(i*i + i*j + j*j) + 24*k*k) * (x^(6 - 12*(i+j) - 8*k) + x^(3 - 3*(i+j) + 16*k)). - Michael Somos, Feb 11 2018

Example

G.f. = 3*x^3 + x^6 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + 2*x^17 + 2*x^18 + 2*x^19 + ...

Mathematica

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a[n_] := SeriesCoefficient[x^3*(f[x^3, x^15]*(f[x^16, x^32]* f[x^15, x^39] + x^6*f[x^8, x^40]*f[x^3, x^51]) + f[x^6, x^12]*(f[x^16, x^32]*f[x^12, x^42] + f[x^8, x^40]*f[x^24, x^30])), {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 02 2018 *)

Crossrefs

Cf. A004012, A005872, A005873, A005874, A005889.

Sequence in context: A204814 A174903 A167163 * A104515 A337923 A363880

Adjacent sequences: A005887 A005888 A005889 * A005891 A005892 A005893

Keyword

nonn

Author

N. J. A. Sloane

Status

approved