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A002325
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Glaisher's J numbers.
(Formerly M0043 N0013)
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14
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1, 1, 2, 1, 0, 2, 0, 1, 3, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 0, 0, 2, 0, 2, 1, 0, 4, 0, 0, 0, 0, 1, 4, 2, 0, 3, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 1, 1, 4, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 0, 1, 0, 4, 2, 2, 0, 0, 0, 3, 2, 0, 2, 2, 0, 0, 0, 0, 5, 2, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 1, 6, 1, 0, 4, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Number of integer solutions to the equation x^2 + 2*y^2 = n when (-x, -y) and (x, y) are counted as the same solution.
For n nonzero, a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
N. J. A. Sloane, Transforms
Index entries for sequences related to Glaisher's numbers
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FORMULA
| Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -2.
Moebius transform is period 8 sequence [ 1, 0, 1, 0, -1, 0, -1, 0, ...]. - Michael Somos, Aug 23 2005
G.f.: (theta_3(q) * theta_3(q^2) - 1) / 2 = Sum_{k>0} kronecker(-8, n) * x^k / (1 - x^k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
Multiplicative with a(2^e) = 1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8). - Michael Somos, Oct 23 2006
A033715(n) = 2 * a(n) unless n=0.
a(n) = A188169(n)+A188170(n)-A188171(n)-A188172(n) [Hirschhorn] - R. J. Mathar, Mar 23 2011
G.f.: A(x)=2*(1+x^2)/(G(0)-2*x*(1+x^2)) ; G(k)=1+x+x^(2*k)*(1+x^3+x^(2*k+1)+x^(2*k+4)+x^(4*k+3)+x^(4*k+4))-x*(1+x^(2*k))*(1+x^(2*k+4))*(1+x^(4*k+4))^2/G(k+1); (continued fraction Euler's kind,1-step ).- Sergei N. Gladkovskii, Jan 03 2012
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EXAMPLE
| x + x^2 + 2*x^3 + x^4 + 2*x^6 + x^8 + 3*x^9 + 2*x^11 + 2*x^12 + x^16 + ...
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MATHEMATICA
| a[n_] := Total[ KroneckerSymbol[-8, #] & /@ Divisors[n]]; Table[a[n], {n, 1, 105}] (* From Jean-François Alcover, Nov 25 2011, after Michael Somos *)
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PROG
| (PARI) a(n) = if( n<1, 0, issquare(n)-issquare(2*n) + 2*sum(i=1, sqrtint(n\2), issquare(n-2*i^2)))
(PARI) {a(n) = if( n<1, 0, qfrep([ 1, 0; 0, 2], n)[n])} /* Michael Somos, Jun 05 2005 */
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker( -2, p) * X))[n])} /* Michael Somos, Jun 05 2005 */
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -8, d)))} /* Michael Somos, Aug 23 2005 */
(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p%8<4, e+1, !(e%2))))))} /* Michael Somos, Oct 23 2006 */
(PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n) / 2)}
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CROSSREFS
| Cf. A033715.
Sequence in context: A036577 * A129134 A133693 A065675 A194313 A127476
Adjacent sequences: A002322 A002323 A002324 * A002326 A002327 A002328
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KEYWORD
| nonn,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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