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 A002173 a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2. (Formerly M4467 N1895) 10
 1, 1, -8, 1, 26, -8, -48, 1, 73, 26, -120, -8, 170, -48, -208, 1, 290, 73, -360, 26, 384, -120, -528, -8, 651, 170, -656, -48, 842, -208, -960, 1, 960, 290, -1248, 73, 1370, -360, -1360, 26, 1682, 384, -1848, -120, 1898, -528, -2208, -8, 2353, 651, -2320, 170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Multiplicative because it is the Inverse Moebius transform of [1, 0, -3^2, 0, 5^2, 0, -7^2, ...], which is multiplicative. - Christian G. Bower, May 18 2005 REFERENCES N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7). N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy] J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8). F. Jarvis, H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171. J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005. FORMULA Multiplicative with a(p^e) = 1 if p = 2; ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4); ((-p^2)^(e+1)-1)/(-p^2-1) if p == 3 (mod 4). - David W. Wilson, Sep 01 2001 [This can be written as a single formula: a(p^e) = ((p^2*Chi(p))^(e+1) - 1)/(p^2*Chi(p) - 1), Chi = A101455. - Jianing Song, Oct 30 2019] G.f.: Sum_{n>=1} A056594(n-1)*n^2*q^n/(1-q^n). Expansion of (1 - theta_4(q)^2 * theta_4(q^2)^4)/4 in powers of q. - Michael Somos, Aug 09 2006 Expansion of (1-eta(q)^4*eta(q^2)^6/eta(q^4)^4)/4 in powers of q. G.f.: qG'(q)/G(q), with G(q) = Prod[n>=1, (1-q^n)^(4n*A056594(n+1)) ]. EXAMPLE The divisors of 15 are 1,3,5,15, so a(15)=(1^2+5^2)-(3^2+15^2) = -208. G.f. = x + x^2 - 8*x^3 + x^4 + 26*x^5 - 8*x^6 - 48*x^7 + x^8 + 73*x^9 + ... - Michael Somos, Jun 25 2019 MATHEMATICA QP = QPochhammer; s = (1-QP[q]^4*(QP[q^2]^6/QP[q^4]^4))/(4*q) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *) a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^2]^4) / 4, {q, 0, n}]; (* Michael Somos, Jun 25 2019 *) PROG (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d)))} /* Michael Somos, Aug 09 2006 */ (Haskell) a002173 n = a050450 n - a050453 n  -- Reinhard Zumkeller, Jun 17 2013 CROSSREFS Equals A050450(n) - A050453(n). A120030(n) = -4*a(n), if n>0. Cf. A056594. Sequence in context: A019432 A211796 A138505 * A050458 A125166 A211785 Adjacent sequences:  A002170 A002171 A002172 * A002174 A002175 A002176 KEYWORD sign,mult,look AUTHOR EXTENSIONS More terms from David W. Wilson STATUS approved

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Last modified September 29 12:15 EDT 2020. Contains 337431 sequences. (Running on oeis4.)