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A103221 Number of partitions of n into parts 2 and 3. 28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.

J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.

Luc Rousseau, a(n) in an hexagonal tiling

T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.

William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))

William A. Stein, The modular forms database

Index entries for Molien series

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

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

EXAMPLE

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 + ...

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 *)

PROG

(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

(Sage) def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # Michael Somos, May 29 2013

(MAGMA) [Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // Vincenzo Librandi, Sep 18 2016

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.

Sequence in context: A011960 A187035 A008615 * A026806 A261348 A320536

Adjacent sequences:  A103218 A103219 A103220 * A103222 A103223 A103224

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Jan 25 2005

EXTENSIONS

Name changed by Wolfdieter Lang, Sep 16 2016

STATUS

approved

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)