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A098108 a(n) = 1 if n is an odd square, otherwise 0. 17
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 = -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
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
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.
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
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);
# 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 *)
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).
Sequence in context: A030217 A030215 A283020 * A363712 A030214 A369658
KEYWORD
nonn,easy,mult
AUTHOR
N. J. A. Sloane, Nov 03 2004
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)