login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A098108 a(n) = 1 if n is an odd square, otherwise 0. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

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

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 Math StackExchange 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

REFERENCES

L. 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.

N. 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

Anonymous, 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

FORMULA

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

EXAMPLE

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

MAPLE

add(x^((m+1/2)^2), m=-10..10);

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 *)

PROG

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

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

CROSSREFS

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

Cf. A000594, A010054.

Sequence in context: A030217 A030215 A283020 * A030214 A025464 A162518

Adjacent sequences:  A098105 A098106 A098107 * A098109 A098110 A098111

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane, Nov 03 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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 21 19:56 EST 2019. Contains 319350 sequences. (Running on oeis4.)