A093709 Characteristic function of squares or twice squares.

1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Apr 11 2004

Keywords

• nonn
• mult

Comments

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

Partial sums of a(n) for n >= 1 are A071860(n+1). - Jaroslav Krizek, Oct 18 2009

For n > 0, this is also the number of different triangular polyabolos that can be formed from n congruent isosceles right triangles (illustrated at A245676). - Douglas J. Durian, Sep 10 2017

Examples

			G.f. = 1 + q + q^2 + q^4 + q^8 + q^9 + q^16 + q^18 + q^25 + q^32 + q^36 + q^49 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
• John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. - From _N. J. A. Sloane_, Feb 23 2009
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for characteristic functions

Crossrefs

Cf. A000203, A000593, A001227, A028982, A053866, A092869.

Programs

Magma

A := Basis( ModularForms( Gamma1(8), 1/2), 104); A[1] + A[2]; /* Michael Somos, Jan 01 2015 */

Maple

seq(`if`(issqr(n) or issqr(n/2),1,0), n=0..100); # Robert Israel, Apr 05 2016

Mathematica

Table[Boole[IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[2*n]]], {n, 0, 104}] (* Jean-François Alcover, Dec 05 2013 *)
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors[ n]] || OddQ [ Length @ Divisors[ 2 n]]]]; (* Michael Somos, Jan 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}]; (* Michael Somos, Jan 01 2015 *)

PARI

{a(n) = issquare(n) || issquare(2*n)};

Formula

Expansion of psi(q^4) * f(-q^3, -q^5) / f(-q, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.

Expansion of f(-q^3, -q^5)^2 / psi(-q) in powers of q where psi(), f() are Ramanujan theta functions. - Michael Somos, Jan 01 2015

Euler transform of period 8 sequence [ 1, 0, -1, 1, -1, 0, 1, -1, ...].

G.f. A(x) satisfies A(x^2) = (A(x) + A(-x)) / 2. a(2*n) = a(n).

Given g.f. A(x), then A(x) / A(x^2) = 1 + x*A092869(x^2).

Given g.f. A(x), then B(x) = A(x^2) / A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + v - 2(u + u^2)*v + 2*(u*v)^2.

Multiplicative with a(0) = a(2^e) = 1, a(p^e) = 1 if e even, 0 otherwise.

a(n) = A053866(n) unless n=0. Characteristic function of A028982 union 0.

G.f.: (theta_3(q) + theta_3(q^2)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(2*k^2)).

Dirichlet g.f.: zeta(2*s) * (1 + 2^-s).

For n>0: a(n) = A010052(n) + A010052(A004526(n))*A059841(n). - Reinhard Zumkeller, Nov 14 2009

a(n) = A000035(A000203(n)) = A000035(A000593(n)) = A000035(A001227(n)), if n>0. - Omar E. Pol, Apr 05 2016

Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020