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 A098108 a(n) = 1 if n is an odd square, otherwise 0. 13
 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 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 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

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Last modified October 23 20:12 EDT 2019. Contains 328373 sequences. (Running on oeis4.)