A008615 a(n) = floor(n/2) - floor(n/3).

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

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

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 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 (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.

It appears that this is also the number of partitions of 2n+6 that are 4-term 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 base-3 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

Number of partitions of (n-2) 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(n-2) = floor((n-2)/2) - floor((n-2)/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(n-2) + [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 n-th row of A340985. That is, the number of weakly connected components of the Collatz digraph of order n. - Sebastian Karlsson, Feb 23 2021

Examples

			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 + ...
		

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, Springer-Verlag, 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 Molien-Weyl, SMF, Vuibert, p. 79

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• David Broadhurst, Feynman integrals, L-series and Kloosterman moments, arXiv:1604.03057 [physics.gen-ph], 2016. See Cor. 1.
• J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, 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.
• A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
• T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 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).

Crossrefs

Essentially the same as A103221.

First differences of A069905 (and A001399).

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

Programs

Haskell

a008615 n = n `div` 2 - n `div` 3  -- Reinhard Zumkeller, Apr 28 2014

Magma

[Floor(n/2)-Floor(n/3): n in [0..10]]; // Sergei Haller (sergei(AT)sergei-haller.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*n-4))>; /* Michael Somos, Dec 11 2018 */

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 *)

PARI

{a(n) = (n\2) - (n\3)}; /* Michael Somos, Feb 06 2003 */

Python

def A008615(n): return n//2 - n//3 # Chai Wah Wu, Jun 07 2022

Formula

a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000

G.f.: x^2 / ((1-x^2) * (1-x^3)).

From Reinhard Zumkeller, Feb 27 2008: (Start)

a(A016933(n)) = a(A016957(n)) = a(A016969(n)) = n+1.

a(A008588(n)) = a(A016921(n)) = a(A016945(n)) = n. (End)

a(6*k) = k, k >= 0. - Zak Seidov, Sep 09 2012

a(n) = A005044(n+1) - A005044(n-3). - 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..n-2} (floor(i/6)-floor((i-3)/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

a(n) = Sum_{k=1..floor((n+3)/2)} Sum_{j=k..floor((2*n+6-k)/3)} Sum_{i=j..floor((2*n+6-j-k)/2)} ([j-k = i-j = 2*n+6-2*i-j-k] - [k = j = i = 2*n+6-i-j-k]), 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*(x-1)*sinh(x))/18. - Stefano Spezia, Oct 17 2022