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Search: id:A008615
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| 0, 0, 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
(list; graph; listen)
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OFFSET
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0,9
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COMMENT
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If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1-x^2)*(1-x^3)), which arises in many contexts. For example, 1/((1-x^4)*(1-x^6)) is the 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).
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 1 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of ways n can be written as the sum of a positive even number and a nonnegative multiple of 3, and so the number of ways (n-2) can be written as the sum of a nonnegative even number and a nonnegative multiple of 3, and also the number of ways (n+3) can be written as the sum of a positive even number and a positive multiple of 3.
a(A016933(n))=a(A016957(n))=a(A016969(n))=n+1; - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 27 2008
a(A008588(n))=a(A016921(n))=a(A016945(n))=n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 27 2008
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REFERENCES
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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
T. Shioda, On the graded ring of invariants of binary octavics. Amer. J. Math. 89, 1022-1046, 1967.
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LINKS
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INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 212
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 448
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
J. Tanton, Young students approach integer triangles
Index entries for Molien series
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FORMULA
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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.: x^2/((1-x^2)*(1-x^3)).
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MAPLE
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[ seq(floor(n/2)-floor(n/3), n=0..100) ];
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PROGRAM
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(PARI) a(n)=(n\2)-(n\3)
(MAGMA) [ Floor(n/2)-Floor(n/3) : n in [0..10]]; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
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CROSSREFS
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First differences of A069905 (and A001399). Essentially the same as A103221.
Sequence in context: A078704 A032358 A011960 this_sequence A103221 A026806 A053280
Adjacent sequences: A008612 A008613 A008614 this_sequence A008616 A008617 A008618
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
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