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A008615
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[n/2] - [n/3].
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34
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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; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| 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
It appears that this is also the number of partitions of 2n that are 4-term arithmetic progressions. [From John W. Layman (layman(AT)math.vt.edu), May 01 2009]
a(n) is the number of (n+3)-digit fixed points under the base-3 Kaprekar map A164993 (see A164997 for the list of fixed points). [From Joseph Myers (jsm(AT)polyomino.org.uk), Sep 04 2009]
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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
T. Shioda, On the graded ring of invariants of binary octavics. Amer. J. Math. 89, 1022-1046, 1967.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for sequences related to linear recurrences with constant coefficients
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
| 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
| [ seq(floor(n/2)-floor(n/3), n=0..100) ];
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MATHEMATICA
| a[n_]:=Floor[n/2]-Floor[n/3]; Array[a, 90, 0] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 05 2008]
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PROG
| (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
| First differences of A069905 (and A001399). Essentially the same as A103221.
Sequence in context: A032358 A011960 A187035 * A103221 A026806 A053280
Adjacent sequences: A008612 A008613 A008614 * A008616 A008617 A008618
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Corrected first Mathematica program [from Harvey P. Dale, Nov 30 2011]
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