This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 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 Cf. A000203, A000593, A001227, A028982, A053866, A092869. Sequence in context: A076141 A011751 A214507 * A079295 A088025 A192687 Adjacent sequences:  A093706 A093707 A093708 * A093710 A093711 A093712 KEYWORD nonn,mult AUTHOR Michael Somos, Apr 11 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 20 20:16 EDT 2019. Contains 323426 sequences. (Running on oeis4.)