

A008615


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


54



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.
It appears that this is also the number of partitions of 2n+6 that are 4term arithmetic progressions.  John W. Layman, May 01 2009 [verified by Wesley Ivan Hurt, Jan 17 2021]
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 at least two of them are equal to each other.  Michael Somos, Mar 01 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
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)
Assuming the Collatz conjecture to be true, for n > 1, a(n+7) is the row length of the nth row of A340985. That is, the number of weakly connected components of the Collatz digraph of order n.  Sebastian Karlsson, Feb 23 2021


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



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(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) = Sum_{i=0..n2} (floor(i/6)floor((i3)/6))*(1)^i.  Wesley Ivan Hurt, Sep 08 2015
a(n) = Sum_{k=1..floor((n+3)/2)} Sum_{j=k..floor((2*n+6k)/3)} Sum_{i=j..floor((2*n+6jk)/2)} ([jk = ij = 2*n+62*ijk]  [k = j = i = 2*n+6ijk]), where [ ] is the (generalized) Iverson bracket.  Wesley Ivan Hurt, Jan 17 2021
E.g.f.: (3*(2 + x)*cosh(x)  2*exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*(x1)*sinh(x))/18.  Stefano Spezia, Oct 17 2022


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

LinearRecurrence[{0, 1, 1, 0, 1}, {0, 0, 1, 0, 1}, 100]; (* Vincenzo Librandi, Sep 09 2015 *)


PROG

(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)
(Python)


CROSSREFS

Cf. A002264, A004009, A004526, A005044, A008588, A008611, A013973, A016921, A016933, A016957, A049310, A057078, A164993, A164997, A340985.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



