William Lionheart - cp's OEIS frontend

User: William Lionheart

William Lionheart has authored 2 sequences.

A307897 Duplicate of A008651.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4

Offset: 0

William Lionheart, May 04 2019

dead

Meyer's generating function h(t,G) generates the sequence of the dimensions of the spaces of G-invariant harmonic polynomials of each degree, where G is a point group on three-dimensional Euclidean space. For G=I, the icosahedral rotation group, the generating function gives rise to this sequence. See Table 1, p. 143.

Burnett Meyer, On the symmetries of spherical harmonics, Canadian Journal of Mathematics 6 (1954): 135-157.

CoefficientList[ Series[(1 - t^10)^(-1) (1 - t^6)^(-1) (1 + t^15), {t, 0, 100}], t]

Vec((1 + x - x^3 - x^4 - x^5 + x^7 + x^8) / ((1 - x)^2*(1 + x)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, May 04 2019

G.f.: (1 + t^15) / ( (1 - t^10) * (1 - t^6) ).

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) - a(n-8) - a(n-9) for n>8. - Colin Barker, May 04 2019

A325488 Dimensions of space of harmonic polynomials of each degree invariant under the full icosahedral group.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 4, 0, 3, 0, 3, 0, 4, 0, 3, 0, 4

Offset: 0

William Lionheart, May 04 2019

Meyer's generating function h(t,G) generates the sequence of the dimensions of the spaces of G-invariant harmonic polynomials of each degree, where G is a point group on three-dimensional Euclidean space. For G=I_h, the full icosahedral group including inversions, the generating function is 1/((1 - t^10)*(1 - t^6)).

Burnett Meyer, On the symmetries of spherical harmonics, Canadian Journal of Mathematics 6 (1954): 135-157.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,-1).

Cf. A008651 for the icosahedral rotation group which is derived from this sequence using Theorem 8 of Meyer, h(t,I)=(1+t^15)*h(t,I_h) as I_h has 15 symmetry planes.

CoefficientList[Series[(1 - t^10)^(-1) (1 - t^6)^(-1) , {t, 0, 100}],
  t]

Vec(1 / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^100)) \\ Colin Barker, Jun 26 2019

G.f.: 1/((1 - t^10)*(1 - t^6)).

a(n) = a(n-6) + a(n-10) - a(n-16) for n>15. - Colin Barker, Jun 26 2019

cp's OEIS Frontend

User: William Lionheart

A307897 Duplicate of A008651.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula