A098108 a(n) = 1 if n is an odd square, otherwise 0.

0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 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, 0, 0, 0, 0, 0, 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, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane, Nov 03 2004

Keywords

• nonn
• easy
• mult

Comments

Motivated by expansion of Jacobi theta function theta_2(x) = Sum_{m = -oo..oo} x^((m+1/2)^2) = 2*Sum_{m odd > 0} q^(m^2/4).

a(n) for n >= 1 is also equal to the Ramanujan number A000594(n) read mod 2. This follows from a theorem started by V. Kumar Murty (2011). Thanks to Benoit Cloitre for this reference. - N. J. A. Sloane, Aug 29 2017

The identification of this sequence with A000594 mod 2 was answered in Mathematics Stack Exchange question 71251. The idea is that (1 - q - q^2 + q^5 + q^7 - ...)^3 = 1 - 3*q + 5*q^3 - 7*q^6 + ... . Reduce mod 2 giving 1 + q + q^3 + q^6 + ... and using (x + y)^2 == (x^2 + y^2) (mod 2) three times gives (1 + q + q^3 + q^6 + ...)^8 == (1 + q^8 + q^24 + q^48 + ...) (mod 2) and we are done. - Michael Somos, Sep 12 2017

Examples

			G.f. = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + q^361 + ...
		

References

• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5n].
• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
• Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.12).
• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65025
• Mathematics Stack Exchange, Converting an infinite product to sum; Ramanujan tau function.
• V. Kumar Murty, The Tau of Ramanujan, Slides of a talk given at the Indian Institute of Science Education and Research, Bhopal, India, Oct 10, 2011. See slide 63/95.
• Ken Ono, Sinai Robins and Patrick T. Wahl, On the Representation of Integers as Sums of Triangular Numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
• H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
• Index entries for characteristic functions.

Crossrefs

Cf. A000122 (theta_3), A002448 (theta_4).

Cf. A000035, A010052, A016754, A000594, A010054.

Programs

Maple

add(x^((m+1/2)^2),m=-10..10);
# alternative
A098108 := proc(n)
    if issqr(n) and type(n,'odd') then
        1;
    else
        0 ;
    end if;
end proc:
seq(A098108(n),n=0..30) ; # R. J. Mathar, Feb 22 2021

Mathematica

Table[If[OddQ@ n && IntegerQ@ Sqrt[n], 1, 0], {n, 0, 120}] (* Michael De Vlieger, Mar 08 2015 *)
Array[Boole@ OddQ@ RamanujanTau@ # &, 120] (* Michael De Vlieger, Aug 27 2017 *)

PARI

{a(n) = n%2 && issquare( n)}; /* Michael Somos, Jun 08 2012 */

PARI

A126811(n) = (ramanujantau(n)%2); \\ Antti Karttunen, Aug 27 2017

Formula

Multiplicative with a(p^e) = 1 if 2 divides e and p > 2, 0 otherwise. - Mitch Harris, Jun 09 2005

Dirichlet g.f.: zeta(2*s)*(1-2^(-2*s)). - R. J. Mathar, Mar 10 2011

G.f.: theta_2( 0, q^4) / 2. - Michael Somos, Jun 08 2012

Euler transform of period 16 sequence [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, ...]. - Michael Somos, Jun 08 2012

a(8*n + 1) = A010054(n). a(n) = 0 unless n == 1 (mod 8). - Michael Somos, Jun 08 2012

a(n) = A000035(n)*A010052(n). - Michel Marcus, Jun 09 2014

For n > 0, a(n) = floor( (sqrt(n)+1)/2 ) - floor( (sqrt(n-1)+1)/2 ). - Mikael Aaltonen, Mar 08 2015

G.f.: eta quotient eta(16*tau)^2/eta(8*tau) = q*Product_{n>=1} (1-q^(16*n))^2 / Product_{n>=1} (1-q^(8*n)), with q = exp(2*Pi*I*z), Im(z) > 0. See the Ono et al. reference, p. 4. - Wolfdieter Lang, Jan 11 2017

Sum_{k=1..n} a(k) ~ sqrt(n)/2. - Amiram Eldar, Oct 28 2023