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A093709 Characteristic function of squares or twice squares. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for characteristic functions

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

EXAMPLE

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

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-Fran├žois 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 *)

PROG

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

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

CROSSREFS

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

Sequence in context: A076141 A011751 A214507 * A079295 A088025 A082416

Adjacent sequences:  A093706 A093707 A093708 * A093710 A093711 A093712

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Apr 11 2004

STATUS

approved

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Last modified February 24 05:37 EST 2018. Contains 299597 sequences. (Running on oeis4.)