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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
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
Eric Weisstein's World of Mathematics, Ramanujan Theta 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
Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020
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-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 *)
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
Sequence in context: A011751 A341540 A214507 * A079295 A088025 A192687
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Apr 11 2004
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)