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.

A307897 Duplicate of A008651.

Original entry on oeis.org

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

Views

Author

William Lionheart, May 04 2019

Keywords

Comments

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.

Links

Programs

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

Formula

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

A087866 Composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type E_8 (binary icosahedral group).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 6, 7, 7, 9, 8, 9, 8, 9, 8, 10, 9, 10, 9, 11, 10, 12, 11, 12, 10, 12, 11, 13, 12, 14, 12, 14, 13, 15, 13, 15, 13, 15, 14, 17, 15, 17, 15, 17, 15, 18, 16, 18, 16, 19, 17, 20, 18, 20, 17, 20, 18, 21, 19, 22, 19
Offset: 0

Views

Author

Paul Boddington, Oct 27 2003

Keywords

References

  • Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.

Links

Crossrefs

Cf. A008651.

Programs

  • Mathematica
    CoefficientList[Series[(1-x^15)/((1-x)(1-x^6)(1-x^10)),{x,0,100}],x] (* Harvey P. Dale, Jan 20 2019 *)
  • PARI
    a(n)=polcoeff((1-x^15)/((1-x)*(1-x^6)*(1-x^10))+O(x^(n+1)),n)

Formula

G.f.: (1-x^15)/((1-x)*(1-x^6)*(1-x^10)).
a(n) = n/60*(15+(-1)^n+b(n)) where b(n) is the 30-periodic sequence {60, 46, 28, 18, -4, -10, 24, 22, -8, -6, 20, 26, 48, 58, 16, -30, -16, 2, 12, 34, 40, 6, 8, 38, 36, 10, 4, -18, -28, 14}. - Benoit Cloitre, Oct 27 2003

Extensions

More terms from Benoit Cloitre, Oct 27 2003

A210576 Positive integers that cannot be expressed as sum of one or more nontrivial binomial coefficients.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29
Offset: 1

Views

Author

Douglas Latimer, Mar 22 2012

Keywords

Comments

The nontrivial binomial coefficients are C(n,k), 2 <= k <= n-2 (A006987).
I conjectured that the sequence is finite, consisting of the terms listed.
This conjecture is now proved. - Douglas Latimer, Apr 10 2013
Note that this sequence allows the same binomial coefficient to be used multiple times. - T. D. Noe, Apr 12 2013
These are the only values of the angular momentum for which a wavefunction with such an angular momentum and the symmetry of a dodecahedron is impossible. [Baez] - Andrey Zabolotskiy, Mar 28 2018

Examples

			The smallest terms in the sequence are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14 because 6, 10 and 15 cannot be terms, as these are the lowest nontrivial binomial coefficients; 12 and 16 cannot be terms, as these are the lowest sums of two nontrivial binomial coefficients; and sums of three or more nontrivial binomial coefficients cannot exclude any of the listed terms.

Links

Crossrefs

A210578 contains many of the integers that cannot be elements of this sequence.
Cf. A006987 and A007318.
Positions of zeros in A008651. Cf. A005796.

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

Views

Author

William Lionheart, May 04 2019

Keywords

Comments

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)).

Links

Crossrefs

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.

Programs

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

Formula

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
Showing 1-4 of 4 results.

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