A214295 a(n) = 1 if n is a square, -1 if n is three times a square, 0 otherwise.

1, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Author

Michael Somos, Jul 10 2012

Keywords

• sign
• mult

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(A092206(n)) = 0; a(A000290(n)) = 1; a(A033428(n)) = -1.

Examples

			G.f. = q - q^3 + q^4 + q^9 - q^12 + q^16 + q^25 - q^27 + q^36 - q^48 + q^49 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 3.
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A000700, A010052, A214293, A244612, A247133, A245485.

Programs

Haskell

a214295 n = a010052 n - a010052 (3*n)  -- Reinhard Zumkeller, Jul 12 2012

Magma

Basis( ModularForms( Gamma1(12), 1/2), 50) [2] ; /* Michael Somos, Jun 10 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}];
a[ n_] := Boole[ IntegerQ[ Sqrt[ n]]] - Boole[ IntegerQ[ Sqrt[ 3 n]]]; (* Michael Somos, Jun 10 2014 *)
Table[Which[IntegerQ[Sqrt[n]],1,IntegerQ[Sqrt[n/3]],-1,True,0],{n,120}] (* Harvey P. Dale, Apr 08 2013 *)

PARI

{a(n) = issquare(n) - issquare(3*n)};

PARI

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 - X, 1) / (1 - X^2 ))[n])};

Formula

Expansion of q * psi(q^3) * f(-q^2, -q^10) / f(-q^5, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.

Multiplicative with a(3^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.

G.f.: (theta_3(q) - theta_3(q^3)) / 2 = Sum_{k>0} x^(k^2) - x^(3*k^2).

Dirichlet g.f.: zeta(2*s) * (1 - 3^(-s)). [corrected by Amiram Eldar, Oct 24 2023]

a(3*n) = - a(n). - Reinhard Zumkeller, Jul 12 2012

Expansion of (phi(q) - phi(q^3)) / 2 = q * chi(q) * f(-q, -q^11) in powers fof q where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 10 2015

Euler transform of period 12 sequence [ 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, -1, ...]. - Michael Somos, Jan 10 2015

Convolution product of A000700 and A247133. - Michael Somos, Jan 10 2015

Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(3) = 0.42264973... . - Amiram Eldar, Oct 24 2023