A195198 Characteristic function of squares or three times squares.
1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 0
Examples
G.f. = 1 + q + q^3 + q^4 + q^9 + q^12 + q^16 + q^25 + q^27 + q^36 + q^48 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. See p. 133 Theorem 3.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
-
Mathematica
a[ n_] := SeriesCoefficient[ Series[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}], {q, 0, n}]; a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n] || OddQ [ Length @ Divisors[3 n]]]]; Table[If[AnyTrue[{Sqrt[n],Sqrt[3n]},IntegerQ],1,0],{n,0,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 22 2020 *) -
PARI
{a(n) = issquare(n) || issquare(3*n)}; -
PARI
{a(n) = if( n<1, n==0, direuler( p=2, n, if( p==3, 1 + X, 1) / (1 - X^2))[n])};
Formula
Euler transform of period 12 sequence [1, -1, 1, 0, 0, -1, 0, 0, 1, -1, 1, -1, ...].
Expansion of psi(q^3) * f(-q^2, -q^10) / f(-q, -q^11) in powers of q where psi(), is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
Multiplicative with a(0) = a(3^e) = 1, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) + theta_3(q^3)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(3*k^2)).
Dirichlet g.f.: zeta(2*s) * (1 + 3^-s).
a(n) = A145377(n) unless n=0. a(3*n + 2) = 0. a(2*n + 1) = A127692(n). a(3*n) = a(n). a(3*n + 1) = A089801(n).
Sum_{k=0..n} a(k) ~ (1+1/sqrt(3)) * sqrt(n). - Amiram Eldar, Sep 14 2023
Comments