cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

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Author

N. J. A. Sloane

Keywords

Comments

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

References

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

Links

Crossrefs

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

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

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).

Examples

			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 + ...

References

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

Links

Crossrefs

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

Programs

  • 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 *)

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

A217511 Theta series of hexagonal diamond or Lonsdaleite net with respect to an atom.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 18
Offset: 0

Views

Author

N. J. A. Sloane, Oct 05 2012

Keywords

Comments

Eq. (20) of [Sloane, 1987] gives the g.f. of this sequence if one replaces the binomial in the round brackets with the factor eta_{3/8}(X^(16/3)); this error propagated from Eq. (67) of [Sloane & Teo, 1985], where the second curly brackets should be replaced by psi_{8/3}(q^(16/3)) to get the g.f. of A005873 (or, alternatively, replace the power 4/3 with 1/3 in both formulas). - Andrey Zabolotskiy, Jun 04 2022

Links

Crossrefs

Cf. A005925, A008264, A004012, A005873.

Formula

a(n) = A004012(n/8) + A005873(n), where the 1st term is 0 unless 8|n. - Andrey Zabolotskiy, Jun 03 2022

Extensions

Missing a(71) = 0 inserted by Andrey Zabolotskiy, Jun 03 2022

A005870 Numbers represented by hexagonal close-packing.

Original entry on oeis.org

0, 3, 6, 8, 9, 11, 12, 15, 17, 18, 19, 20, 21, 22, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 61, 63, 65, 66, 67, 68, 69, 70, 71, 72, 75
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers n such that A004012(n) != 0. - Sean A. Irvine, Sep 23 2016

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
  • S. E. Thiel, Generating Functions for Fibonomial Coefficients and Fibonacci Products, http://stuartthiel.homestead.com/files/fibonacci/THIEL_fibonomial_coefficients_rev110612.pdf [Broken link?]
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A004012.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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