

A008615


a(n) = floor(n/2)  floor(n/3).


49



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
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OFFSET

0,9


COMMENTS

If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1x^2)*(1x^3)), which arises in many contexts. For example, 1/((1x^4)*(1x^6)) is the 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+8 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 (n2) 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, Feb 27 2008
a(A008588(n)) = a(A016921(n)) = a(A016945(n)) = n.  Reinhard Zumkeller, Feb 27 2008
It appears that this is also the number of partitions of 2n that are 4term arithmetic progressions.  John W. Layman, May 01 2009
a(n) is the number of (n+3)digit fixed points under the base3 Kaprekar map A164993 (see A164997 for the list of fixed points).  Joseph Myers, Sep 04 2009
Starting from n=10 also the number of balls in new consecutive hexagonal edges, if an (infinite) chain of balls is winded spirally around the first ball at the center, such that each six steps make an entire winding.  K. G. Stier, Dec 21 2012
In any three consecutive terms exactly two of them are equal to each other.  Michael Somos, Mar 01 2014
Number of partitions of (n2) into parts 2 and 3.  David Neil McGrath, Sep 05 2014
a(n), n >= 0, is also the dimension of S_{2*(n+4)}, the complex vector space of modular cusp forms of weight 2*(n+4) and level 1 (full modular group). The dimension of S_0, S_2, S_4 and S_6 is 0. See, e.g., Ash and Gross, p. 178. Table 13.1.  Wolfdieter Lang, Sep 16 2016
From Wolfdieter Lang, May 08 2017: (Start)
a(n2) = floor((n2)/2)  floor((n2)/3) = floor(n/2)  floor((n+1)/3) is for n >=0 the number of integers k in the interval (n+1)/3 < k <= floor(n/2). This problem appears in the computation of the number of zeros of Chebyshev S(n, x) polynomials (coefficients in A049310) in the open interval (1, +1). See a comment there. This computation was motivated by a conjecture given in A008611 by Michel Lagneau, Mar 31 2017.
a(n) is also the number of integers k in the closed interval (n+1)/3 <= k <= floor(n/2), which is floor(n/2)  (ceiling((n+1)/3)  1) for n >= 0 (proof trivial for n+1 == 0 (mod 3) and otherwise). From the preceding statement this a(n) is also a(n2) + [n == 2 (mod 3)] for n >= 0 (with [statement] = 1 if the statement is true and zero otherwise). This proves the recurrence given by Michael Somos in the formula section. (End)


REFERENCES

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, SpringerVerlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J.M. Kantor, Où en sont les mathématiques, La formule de MolienWeyl, SMF, Vuibert, p. 79


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
David Broadhurst, Feynman integrals, Lseries and Kloosterman moments, arXiv:1604.03057 [physics.genph], 2016. See Cor. 1.
J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392412, esp. p. 402.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 212
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 448
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 10221046, 1967.
William A. Stein, The modular forms database
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
James Tanton, Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4  6.
James Tanton et al., Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4  6.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,1)


FORMULA

a(n) = a(n6) + 1 = a(n2) + a(n3)  a(n5).  Henry Bottomley, Sep 02 2000
G.f.: x^2 / ((1x^2) * (1x^3)).
a(6*k) = k, k >= 0.  Zak Seidov, Sep 09 2012
a(n) = A005044(n+1)  A005044(n3).  Johannes W. Meijer, Oct 18 2013
a(n) = floor((n+4)/6)  floor((n+3)/6) + floor((n+2)/6).  Mircea Merca, Nov 27 2013
Euler transform of length 3 sequence [0, 1, 1].  Michael Somos, Mar 01 2014
a(n+2) = a(n) + 1 if n == 0 (mod 3), a(n+2) = a(n) otherwise.  Michael Somos, Mar 01 2014. See the May 08 2017 comment above.  Wolfdieter Lang, May 08 2017
a(n) = a(1  n) for all n in Z.  Michael Somos, Mar 01 2014.
a(n) = A004526(n)  A002264(n).  Reinhard Zumkeller, Apr 28 2014
a(n) = Sum_{i=0..n2} (floor(i/6)floor((i3)/6))*(1)^i.  Wesley Ivan Hurt, Sep 08 2015
a(n) = a(n+6)  1 = A103221(n+4)  1, n >= 0.  Wolfdieter Lang, Sep 16 2016
12*a(n) = 2*n +1 +3*(1)^n 4*A057078(n).  R. J. Mathar, Jun 19 2019


EXAMPLE

G.f. = x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + ...


MAPLE

a := n> floor(n/2)  floor(n/3): seq(a(n), n = 0 .. 87);


MATHEMATICA

a[n_]:=Floor[n/2]Floor[n/3]; Array[a, 90, 0] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008; corrected by Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{0, 1, 1, 0, 1}, {0, 0, 1, 0, 1}, 100]; // Vincenzo Librandi, Sep 09 2015


PROG

(PARI) {a(n) = (n\2)  (n\3)}; /* Michael Somos, Feb 06 2003 */
(MAGMA) [Floor(n/2)Floor(n/3): n in [0..10]]; // Sergei Haller (sergei(AT)sergeihaller.de), Dec 21 2006
(MAGMA) a := func< n  n lt 2 select 0 else n eq 2 select 1 else Dimension( ModularForms( PSL2( Integers()), 2*n4))>; /* Michael Somos, Dec 11 2018 */
(Haskell)
a008615 n = n `div` 2  n `div` 3  Reinhard Zumkeller, Apr 28 2014


CROSSREFS

Essentially the same as A103221.
First differences of A069905 (and A001399).
Sequence in context: A032358 A011960 A187035 * A103221 A026806 A261348
Adjacent sequences: A008612 A008613 A008614 * A008616 A008617 A008618


KEYWORD

nonn,easy,nice,changed


AUTHOR

N. J. A. Sloane, Simon Plouffe


STATUS

approved



