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.

A103221 Number of partitions of n into parts 2 and 3.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14, 15, 15
Offset: 0

Views

Author

Michael Somos, Jan 25 2005

Keywords

Comments

Essentially the same as A008615.
Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+12 cusp forms for Gamma_0( 1 ).
Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of partitions of n into two nonnegative parts congruent modulo 3. - Andrew Baxter, Jun 28 2006
Also number of equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008
a(n) is also the number of 2-regular multigraphs on n vertices, where each component is either a pair of parallel edges, or a triangle. - Jason Kimberley, Oct 14 2011
For n>1, a(n) is the number of partitions of 2n into positive parts x,y, and z such that x>=y and y=z. This sequence is used in calculating the probability of the need for a run-off election when n voters randomly cast ballots for two of three candidates running for two empty slots on a county commission. - Dennis P. Walsh, Apr 25 2013
Also, Molien series for invariants of finite Coxeter group A_2. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k, not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016
The coefficient of x^(2*n+1) in the power series expansion of the Weierstrass sigma function is a polynomial in the invariants g2 and g3 with a(n) terms. - Michael Somos, Jun 14 2016
a(n) is also the dimension of the complex vector space of modular forms M_{2*n} of weight 2*n and level 1 (full modular group). See Apostol p. 119, eq. (9) for k=2*n, and Ash and Gross, p. 178, Table 13.1. For a(6*k+1) = a(6*k+j)-1 for j = 0,2,3,4,5 and k >= 0 see A016921 (so-called dips, cf. Ash and Gross, p. 178.). - Wolfdieter Lang, Sep 16 2016
In an hexagonal tiling of the plane where the base tile is (0,0)--(2,1)--(3,3)--(1,4)--(-1,3)--(-2,1)--(0,0), a(n) is the number of vertices on the (n,0)--(n,n) closed line segment. - Luc Rousseau, Mar 22 2018

Examples

			For n=8, a(n)=2 since there are two partitions of 16 into 3 positive parts x, y, and z such that x >= y and y=z, namely, 16 = 8+4+4 and 16 = 6+5+5. - _Dennis P. Walsh_, Apr 25 2013
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...

References

  • T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 119.
  • Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
  • D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
  • E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
  • R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
  • J.-M. Kantor, Ou en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
  • S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 26. - N. J. A. Sloane, Aug 28 2010.

Links

Crossrefs

Cf. A008615, A001399 (partial sums), A128115, A171386, A081753.
Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776, A266777, A266778, A266779, A266780, A266781.
Cf. A004009, A013973, A016921, A079978.

Programs

  • Magma
    [Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // Vincenzo Librandi, Sep 18 2016
  • Maple
    A103221:=n->floor((n+2)/2)-floor((n+2)/3): # Andrew Baxter, Jun 06 2008
  • Mathematica
    a=b=c=d=0;Table[e=a+b-d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1},{1, 0, 1, 1, 1},88] (* Ray Chandler, Sep 23 2015 *)
a[ n_] := With[{m = Max[-5 - n, n]}, (-1)^Boole[n < 0] SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Jun 02 2019 *)
  • PARI
    {a(n) = if( n<-4, -a(-5-n), polcoeff( 1 / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};
  • PARI
    a(n)=n+=2; n\2 - n\3 \\ Charles R Greathouse IV, Jul 31 2017
  • Python
    def A103221(n): return (n>>1)+1-(n+2)//3 # Chai Wah Wu, Apr 15 2025
  • Sage
    def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # Michael Somos, May 29 2013

Formula

Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
a(n) = A008615(n+2). First differences of A001399.
a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: 1/((1-x^2)*(1-x^3)).
a(n) = floor((n+2)/2) - floor((n+2)/3). - Andrew Baxter, Jun 06 2008
For odd n, a(n)=floor((n+3)/6). For even n, a(n)=floor((n+6)/6). - Dennis P. Walsh, Apr 25 2013
a(n) = floor(n/6)+1 unless n == 1 (mod 6); if n == 1 (mod 6), a(n) = floor(n/6). - Bob Selcoe, Sep 27 2014
a(n) = A081753(2*n); see the Dennis P. Walsh formula. - Wolfdieter Lang, Sep 16 2016
a(n)-a(n-2) = A079978(n). - R. J. Mathar, Jun 23 2021
E.g.f.: (3*(4 + x)*cosh(x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)) + 3*(1 + x)*sinh(x))/18. - Stefano Spezia, Mar 05 2023
a(n) = A008615(n-1)+A059841(n). - R. J. Mathar, May 03 2023

Extensions

Name changed by Wolfdieter Lang, Sep 16 2016

A128114 Number of uniform polyhedra with n vertices.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 2, 0, 5, 0, 10, 0, 7, 0, 5, 0, 7, 0, 9, 0, 12, 0, 15, 0, 14, 0, 7, 0, 18, 0, 9, 0, 19, 0, 7, 0, 21, 0, 10, 0, 14, 0, 11, 0, 26, 0, 12, 0, 23, 0, 14, 0, 21, 0, 14, 0, 33, 0, 42, 0, 35, 0, 19, 0, 23, 0, 18, 0, 28, 0, 14, 0, 42, 0, 21, 0, 28, 0, 18, 0, 47, 0, 14, 0, 49, 0, 24, 0
Offset: 1

Views

Author

Paulo de A. Sachs (sachs6(AT)yahoo.de), Feb 15 2007, corrected Feb 15 2007

Keywords

Examples

			Every other term is zero because there are no uniform polyhedra with an odd number of vertices. a(6)=3 because there are the octahedron, the tetrahemihexahedron and the triangular prism.

Links

Crossrefs

Cf. A128914.

Formula

After 120th term, a(n) equals 0 for n odd and 2 * A055684(n/2) + A128115(n/2) + 2 for n even.

A128112 Number of uniform polyhedra with n faces.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 3, 4, 3, 10, 5, 7, 6, 7, 4, 11, 8, 12, 9, 8, 6, 16, 11, 13, 10, 14, 9, 10, 14, 19, 15, 13, 10, 19, 12, 11, 18, 21, 12, 16, 20, 20, 21, 16, 12, 26, 23, 14, 21, 25, 16, 22, 26, 21, 20, 20, 18, 33, 29, 18, 30, 35, 18, 27, 24, 23, 33, 26, 22, 28, 35, 20, 36, 42
Offset: 1

Views

Author

Paulo de A. Sachs (sachs6(AT)yahoo.de), Feb 15 2007, corrected Feb 15 2007

Keywords

Examples

			a(20)=12 because there are the icosahedron, the small cubicuboctahedron, the great cubicuboctahedron, the cubitruncated cuboctahedron, the great icosahedron, the octadecagonal prism, two octadecagrammic (18/5 and 18/7) prisms, the enneagonal antiprism, two enneagrammic (9/2 and 9/4) antiprisms and the enneagrammic crossed antiprism.

Links

Crossrefs

Cf. A128113, A128114.

Formula

After 124th term, a(n) equals A055684(n-2) + 1 for n odd and A055684(n-2) + A055684(n/2-1) + A128115(n/2-1) + 2 for n even.

A128113 Number of uniform polyhedra with n edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 2, 1, 0, 2, 0, 3, 3, 0, 0, 6, 0, 0, 3, 4, 0, 8, 0, 3, 5, 0, 0, 9, 0, 0, 6, 2, 0, 3, 0, 7, 4, 0, 0, 13, 0, 0, 8, 8, 0, 3, 0, 4, 9, 0, 0, 22, 0, 0, 6, 5, 0, 5, 0, 11, 11, 0, 0, 11, 0, 0, 10, 12, 0, 6, 0, 6, 9, 0, 0, 14, 0, 0, 14, 6, 0, 10, 0, 15, 15, 0, 0, 13, 0, 0
Offset: 1

Views

Author

Paulo de A. Sachs (sachs6(AT)yahoo.de), Feb 15 2007, corrected Feb 15 2007

Keywords

Examples

			The first nonzero term, a(6)=1, represents the polyhedron with least edges: the tetrahedron. There is no polyhedron with 7 edges and no polyhedron with 8 edges is uniform, a(9)=1 represents the triangular prism, the next nonzero term, a(12), is 3 because there are the tetrahemihexahedron, the cube and the octahedron.

Links

Crossrefs

Cf. A128112, A128114.

Formula

After 240th term, a(n) equals the sum between [A055684(n/3) + 1 for n != 0 mod 3, otherwise 0] and [A055684(n/4) + A128115(n/4) + 1 for n != 0 mod 4, otherwise 0].
Showing 1-4 of 4 results.

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