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A093709
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Characteristic function of squares or twice squares.
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13
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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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).
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EXAMPLE
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G.f. = 1 + q + q^2 + q^4 + q^8 + q^9 + q^16 + q^18 + q^25 + q^32 + q^36 + q^49 + ...
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MAPLE
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seq(`if`(issqr(n) or issqr(n/2), 1, 0), n=0..100); # Robert Israel, Apr 05 2016
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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