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A098108
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a(n) = 1 if n is an odd square, otherwise 0.
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15
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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G.f. = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + q^361 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = n%2 && issquare( n)}; /* Michael Somos, Jun 08 2012 */
(PARI) A126811(n) = (ramanujantau(n)%2); \\ Antti Karttunen, Aug 27 2017
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CROSSREFS
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Cf. A000122 (theta_3), A002448 (theta_4).
Cf. A016754, A000594, A010054.
Sequence in context: A030217 A030215 A283020 * A030214 A025464 A162518
Adjacent sequences: A098105 A098106 A098107 * A098109 A098110 A098111
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KEYWORD
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nonn,mult
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AUTHOR
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N. J. A. Sloane, Nov 03 2004
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STATUS
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approved
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