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%I #105 May 03 2023 11:17:58
%S 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,
%T 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,
%U 11,11,12,11,12,12,12,12,13,12,13,13,13,13,14,13,14,14,14,14,15,14,15,15
%N Number of partitions of n into parts 2 and 3.
%C Essentially the same as A008615.
%C 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).
%C Dimension of the space of weight 2n+12 cusp forms for Gamma_0( 1 ).
%C Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
%C a(n) is the number of partitions of n into two nonnegative parts congruent modulo 3. - _Andrew Baxter_, Jun 28 2006
%C Also number of equivalence classes of period 2n billiards on an equilateral triangle. - _Andrew Baxter_, Jun 06 2008
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%D T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 119.
%D Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%D E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
%D R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
%D J.-M. Kantor, Ou en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 26. - _N. J. A. Sloane_, Aug 28 2010.
%H Seiichi Manyama, <a href="/A103221/b103221.txt">Table of n, a(n) for n = 0..10000</a>
%H Andrew M. Baxter and Ron Umble, <a href="http://arXiv.org/abs/math/0509292">Periodic Orbits of Billiards on an Equilateral Triangle</a>, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.
%H J. Igusa, <a href="http://www.jstor.org/stable/2373172">On Siegel modular forms of genus 2 (II)</a>, Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
%H Luc Rousseau, <a href="/A103221/a103221.png">a(n) in an hexagonal tiling</a>
%H T. Shioda, <a href="http://www.jstor.org/stable/2373415">On the graded ring of invariants of binary octavics</a>, Amer. J. Math. 89, 1022-1046, 1967.
%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>
%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,-1).
%F Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
%F a(n) = A008615(n+2). First differences of A001399.
%F a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - _Henry Bottomley_, Sep 02 2000
%F G.f.: 1/((1-x^2)*(1-x^3)).
%F a(n) = floor((n+2)/2) - floor((n+2)/3). - _Andrew Baxter_, Jun 06 2008
%F For odd n, a(n)=floor((n+3)/6). For even n, a(n)=floor((n+6)/6). - _Dennis P. Walsh_, Apr 25 2013
%F 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
%F a(n) = A081753(2*n); see the _Dennis P. Walsh_ formula. - _Wolfdieter Lang_, Sep 16 2016
%F a(n)-a(n-2) = A079978(n). - _R. J. Mathar_, Jun 23 2021
%F 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
%F a(n) = A008615(n-1)+A059841(n). - _R. J. Mathar_, May 03 2023
%e 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
%e 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 + ...
%p A103221:=n->floor((n+2)/2)-floor((n+2)/3): # _Andrew Baxter_, Jun 06 2008
%t 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 *)
%t LinearRecurrence[{0, 1, 1, 0, -1},{1, 0, 1, 1, 1},88] (* _Ray Chandler_, Sep 23 2015 *)
%t 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 *)
%o (PARI) {a(n) = if( n<-4, -a(-5-n), polcoeff( 1 / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};
%o (PARI) a(n)=n+=2; n\2 - n\3 \\ _Charles R Greathouse IV_, Jul 31 2017
%o (Sage) def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # _Michael Somos_, May 29 2013
%o (Magma) [Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // _Vincenzo Librandi_, Sep 18 2016
%Y Cf. A008615, A001399 (partial sums), A128115, A171386, A081753.
%Y 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.
%Y Cf. A004009, A013973, A016921, A079978.
%K nonn,easy
%O 0,7
%A _Michael Somos_, Jan 25 2005
%E Name changed by _Wolfdieter Lang_, Sep 16 2016
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