A005889 -id:A005889 - cp's OEIS frontend

A045832 a(n) = A005889(n)/3.

1, 0, 2, 1, 0, 2, 2, 0, 6, 0, 0, 2, 2, 0, 8, 1, 0, 2, 4, 0, 8, 2, 0, 2, 5, 0, 6, 2, 0, 4, 6, 0, 8, 2, 0, 6, 6, 0, 12, 0, 0, 0, 4, 0, 12, 4, 0, 2, 7, 0, 16, 2, 0, 6, 4, 0, 12, 0, 0, 8, 6, 0, 12, 1, 0, 4

Offset: 1

N. J. A. Sloane

nonn

A263433 Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 6, 4, 7, 4, 6, 8, 4, 10, 8, 12, 8, 6, 14, 8, 11, 6, 8, 8, 8, 14, 6, 12, 15, 14, 14, 8, 12, 14, 12, 16, 8, 10, 14, 16, 16, 12, 12, 12, 16, 10, 10, 8, 19, 20, 20, 8, 12, 24, 14, 24, 12, 16, 14, 16, 21, 10, 14, 28, 16, 12, 14, 12, 16, 16, 30, 12

Offset: 0

Michael Somos, Oct 18 2015

nonn

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

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 4*x^7 + 7*x^8 + ...
G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 6*q^37 + 4*q^43 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A005889, A045832, A261426, A263444.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 4, 0, x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A)^4 / (eta(x + A)^2 * eta(x^12 + A)^2), n))};

Expansion of f(-x^2)^2 * phi(-x^6)^2 / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^6)^4 / (eta(q)^2 * eta(q^12)^2) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263444.
a(n) = A261426(2*n) = A045832(6*n). 3 * a(n) = A005889(6*n).

A005874 Theta series of hexagonal close-packing with respect to triangle between tetrahedra.

Original entry on oeis.org

0, 3, 2, 0, 3, 12, 0, 6, 0, 6, 0, 12, 6, 6, 12, 12, 3, 0, 2, 6, 0, 24, 0, 24, 6, 3, 0, 24, 6, 12, 12, 6, 0, 12, 0, 0, 18, 6, 12, 48, 0, 24, 0, 6, 0, 36, 0, 0, 6, 9, 14, 24, 6, 12, 12, 0, 0, 48, 0, 36, 24, 6, 12, 12, 3, 24, 12, 6, 0, 24, 0, 24, 6, 12, 0, 48, 12

Offset: 0

N. J. A. Sloane

Just take the theta series for the h.c.p. and subtract the coordinates of the center of the triangle from each point. - N. J. A. Sloane, May 18 2021

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

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.
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.

Cf. A004012, A005872, A005873, A005889, A005890.

Sum_{n<=x} a(n)^2 ~ (8*Pi^4/(21*zeta(3))) * x^2. (Choi/Kumchev/Osburn) [Corrected by Vaclav Kotesovec, Oct 25 2015]

Terms a(63) and beyond from Andrey Zabolotskiy, Jun 20 2022

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

Original entry on oeis.org

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

N. J. A. Sloane

nonn

The triangle separates a tetrahedron and an octahedron.

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

			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 + ...

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

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

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

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 *)

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

cp's OEIS Frontend

A045832 a(n) = A005889(n)/3.

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A263433 Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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A005874 Theta series of hexagonal close-packing with respect to triangle between tetrahedra.

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

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