A162397 a(n) = n * Kronecker(-3, n).

1, -2, 0, 4, -5, 0, 7, -8, 0, 10, -11, 0, 13, -14, 0, 16, -17, 0, 19, -20, 0, 22, -23, 0, 25, -26, 0, 28, -29, 0, 31, -32, 0, 34, -35, 0, 37, -38, 0, 40, -41, 0, 43, -44, 0, 46, -47, 0, 49, -50, 0, 52, -53, 0, 55, -56, 0, 58, -59, 0, 61, -62, 0, 64, -65, 0

Offset: 1

Author

Michael Somos, Jul 02 2009

Keywords

• sign
• easy
• mult

Comments

In Gil and Robins 2003 on page 33 the g.f. is denoted by f_{4, 2}(x). - Michael Somos, Sep 04 2015

Examples

			G.f. = x - 2*x^2 + 4*x^4 - 5*x^5 + 7*x^7 - 8*x^8 + 10*x^10 - 11*x^11 + ...
		

References

• George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 319, Equation (14.3.6).

Links

• Juan B. Gil and Sinai Robins, Hecke operators on rational functions, arXiv:math/0309244 [math.NT], 2003.

Crossrefs

Cf. A016777, A016789, A091684, A102283.

Programs

Mathematica

Table[n*KroneckerSymbol[-3,n],{n,80}] (* Harvey P. Dale, Mar 14 2015 *)

PARI

{a(n) = n * kronecker(-3, n)};

Formula

Euler transform of length 3 sequence [ -2, -1, 2].

a(n) is completely multiplicative with a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 3), a(p^e) = (-p)^e if p == 2 (mod 3).

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

a(3*n) = 0. a(n) = a(-n). abs(a(n)) = A091684(n). a(n) = n * A102283(n). a(3*n + 1) = A016777(n). a(3*n + 2) = - A016789(n). - Michael Somos, Mar 14 2012

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/3. - Amiram Eldar, Nov 23 2023