A081753 a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.

1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9

Offset: 0

Author

Benoit Cloitre, Apr 08 2003

Keywords

• nonn
• easy

Comments

a(2n) = dimension of M(2n), where M(2n) denotes the complex vector space of modular forms of weight 2n for the group : PSL2(Z). dimension of M(2n+1) = 0.

See A103221(n) for the dimension of M(2n). The Apostol reference, p. 119, eq. (9) uses even k. - Wolfdieter Lang, Sep 16 2016

The space of modular forms is generated by E_4 (A004009) and E_6 (A013973) both of even weight. This is why the space of modular forms of odd weight is trivial. - Michael Somos, Dec 11 2018

Examples

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + ... - _Michael Somos_, Dec 11 2018
		

References

• Apostol, Tom M., Modular Functions and Dirichlet Series in Number Theory, second edition, Springer, 1990.
• Yves Hellegouarch, "Invitation aux mathématiques de Fermat-Wiles", Dunod, 2ème édition, p. 285

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A008615, A097992, A103221.

Programs

Maple

seq(floor(n/12)+1-charfcn[0](n-2 mod 12), n=0..100); # Robert Israel, Sep 16 2016

Mathematica

Table[If[Mod[n, 12] == 2, Floor[n/12], Floor[n/12] + 1], {n, 0, 120}] (* or *)
CoefficientList[Series[(1 - x^2 + x^3)/(1 - x - x^12 + x^13), {x, 0, 120}], x] (* Michael De Vlieger, Sep 19 2016 *)
a[ n_] := Quotient[n, 12] + Boole[Mod[n, 12] != 2]; (* Michael Somos, Dec 11 2018 *)

PARI

a(k) = if(k%12-2, floor(k/12)+1, floor(k/12))

PARI

{a(n) = n\12 + (n%12!=2)}; /* Michael Somos, Dec 11 2018 */

Formula

a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.

G.f.: (1-x^2+x^3)/(1-x-x^12+x^13). - Robert Israel, Sep 16 2016

a(2*n) = A008615(n+2), a(2*n+1) = A097992(n). - Michael Somos, Dec 11 2018