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A103221
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Number of partitions of n with parts of size two and three.
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11
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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; internal format)
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OFFSET
| 0,7
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COMMENTS
| Essentially the same as A008615.
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 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 (baxter(AT)math.rutgers.edu), 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
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REFERENCES
| 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. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
J.-M. Kantor, Ou en sont les mathematiques, 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.
T. Shioda, On the graded ring of invariants of binary octavics. Amer. J. Math. 89, 1022-1046, 1967.
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LINKS
| Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.
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
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FORMULA
| Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
a(n) = a(n-6)+1 = a(n-2)+a(n-3)-a(n-5) - Henry Bottomley (se16(AT)btinternet.com), Sep 02 2000
G.f.: 1/((1-x^2)*(1-x^3)).
a(n) = floor((n+2)/2) - floor((n+2)/3) - Andrew Baxter (baxter(AT)math.rutgers.edu), Jun 06 2008
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MAPLE
| A103221:=n->floor((n+2)/2)-floor((n+2)/3): - Andrew Baxter (baxter(AT)math.rutgers.edu), Jun 06 2008
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MATHEMATICA
| a=b=c=d=0; Table[e=a+b-d+1; a=b; b=c; c=d; d=e, {n, 100}] (*From Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)
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PROG
| (PARI) {a(n)=if(n<-4, -a(-5-n), polcoeff( 1/(1-x^2)/(1-x^3)+x*O(x^n), n))}
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CROSSREFS
| Cf. A008615(n)=a(n-2). First differences of A001399.
Cf. A128115.
Sequence in context: A011960 A187035 A008615 * A026806 A053280 A025832
Adjacent sequences: A103218 A103219 A103220 * A103222 A103223 A103224
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Jan 25 2005
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