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%I #53 Sep 08 2022 08:45:13
%S 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,
%T 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,
%U 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
%N Characteristic function of squares or twice squares.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Partial sums of a(n) for n >= 1 are A071860(n+1). - _Jaroslav Krizek_, Oct 18 2009
%C 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
%H Robert Israel, <a href="/A093709/b093709.txt">Table of n, a(n) for n = 0..10000</a>
%H S. Cooper and M. Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
%H John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. - From _N. J. A. Sloane_, Feb 23 2009
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F 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.
%F 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
%F Euler transform of period 8 sequence [ 1, 0, -1, 1, -1, 0, 1, -1, ...].
%F G.f. A(x) satisfies A(x^2) = (A(x) + A(-x)) / 2. a(2*n) = a(n).
%F Given g.f. A(x), then A(x) / A(x^2) = 1 + x*A092869(x^2).
%F 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.
%F Multiplicative with a(0) = a(2^e) = 1, a(p^e) = 1 if e even, 0 otherwise.
%F a(n) = A053866(n) unless n=0. Characteristic function of A028982 union 0.
%F G.f.: (theta_3(q) + theta_3(q^2)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(2*k^2)).
%F Dirichlet g.f.: zeta(2*s) * (1 + 2^-s).
%F For n>0: a(n) = A010052(n) + A010052(A004526(n))*A059841(n). - _Reinhard Zumkeller_, Nov 14 2009
%F a(n) = A000035(A000203(n)) = A000035(A000593(n)) = A000035(A001227(n)), if n>0. - _Omar E. Pol_, Apr 05 2016
%F Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - _Vaclav Kotesovec_, Oct 16 2020
%e G.f. = 1 + q + q^2 + q^4 + q^8 + q^9 + q^16 + q^18 + q^25 + q^32 + q^36 + q^49 + ...
%p seq(`if`(issqr(n) or issqr(n/2),1,0), n=0..100); # _Robert Israel_, Apr 05 2016
%t Table[Boole[IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[2*n]]], {n, 0, 104}] (* _Jean-François Alcover_, Dec 05 2013 *)
%t a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors[ n]] || OddQ [ Length @ Divisors[ 2 n]]]]; (* _Michael Somos_, Jan 01 2015 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}]; (* _Michael Somos_, Jan 01 2015 *)
%o (PARI) {a(n) = issquare(n) || issquare(2*n)};
%o (Magma) A := Basis( ModularForms( Gamma1(8), 1/2), 104); A[1] + A[2]; /* _Michael Somos_, Jan 01 2015 */
%Y Cf. A000203, A000593, A001227, A028982, A053866, A092869.
%K nonn,mult
%O 0,1
%A _Michael Somos_, Apr 11 2004
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