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%I A002654 M0012 N0001
%S A002654 1,1,0,1,2,0,0,1,1,2,0,0,2,0,0,1,2,1,0,2,0,0,0,0,3,2,0,0,2,0,0,1,0,2,0,
%T A002654 1,2,0,0,2,2,0,0,0,2,0,0,0,1,3,0,2,2,0,0,0,0,2,0,0,2,0,0,1,4,0,0,2,0,0,
%U A002654 0,1,2,2,0,0,0,0,0,2,1,2,0,0,4,0,0,0,2,2,0,0,0,0,0,0,2,1,0,3,2,0,0,2,0
%N A002654 Number of ways of writing n as a sum of at most two nonzero squares, 
               where order matters; also (number of divisors of n of form 4m+1) 
               - (number of divisors of form 4m+3).
%C A002654 Number of sublattices of Z X Z of index n that are similar to Z X Z; 
               number of (principal) ideals of Z[i] of norm n.
%C A002654 G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=(u-v)^2-(v-w)(4w+1). 
               - Michael Somos, Jul 19 2004
%C A002654 a(A022544(n)) = 0; a(A001481(n)) > 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 27 2008]
%D A002654 M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., 
               Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
%D A002654 G. Chrystal, Algebra: An elementary text-book for the higher classes 
               of secondary schools and for colleges, 6th ed, Chelsea Publishing 
               Co., New York 1959 Part II, p. 346 Exercise XXI(17). MR0121327 (22 
               #12066)
%D A002654 J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure 
               and Applied Mathematics, 20 (1884), 97-167.
%D A002654 J. W. L. Glaisher, On the function which denotes the difference between 
               the number of (4m+1)-divisors and the number of (4m+3)-divisors of 
               a number, Proc. London Math. Soc., 15 (1884), 104-122.
%D A002654 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, 
               NY, 1985, p. 15.
%D A002654 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi 
               elliptic functions, continued fractions and Schur functions, Ramanujan 
               J., 6 (2002), 7-149.
%D A002654 John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of 
               plane sublattices by parent Patterson symmetry and colour lattice 
               group type, Acta Cryst. (2009). A65, 156163. [See Table 1]. [From 
               N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]
%D A002654 G. Scheja and U. Storch, Lehrbuch der Algebra, Tuebner, 1988, p. 251.
%D A002654 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002654 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002654 T. D. Noe, <a href="b002654.txt">Table of n, a(n) for n = 1..10000</a>
%H A002654 M. Baake and U. Grimm, <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-392">
               Quasicrystalline combinatorics</a>
%H A002654 <a href="Sindx_Su.html#ssq">Index entries for sequences related to sums 
               of squares</a>
%H A002654 <a href="Sindx_Su.html#sublatts">Index entries for sequences related 
               to sublattices</a>
%H A002654 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002654 Dirichlet series: (1-2^(-s))^(-1)*Product (1-p^(-s))^(-2) (p=1 mod 4) 
               * Product (1-p^(-2s))^(-1) (p=3 mod 4) = Dedekind eta-function of 
               Z[ i ].
%F A002654 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, 
               p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -16.
%F A002654 If n=2^k*u*v, where u is product of primes 4m+1, v is product of primes 
               4m+3, then a(n)=0 unless v is a square, in which case a(n) = number 
               of divisors of u (Jacobi).
%F A002654 Multiplicative with a(p^e) = 1 if p = 2; e+1 if p == 1 (mod 4); (e+1) 
               mod 2 if p == 3 (mod 4).. - David W. Wilson, Sep 01, 2001
%F A002654 G.f.: Sum((-1)^floor(n/2)*x^((n^2+n)/2)/(1+(-x)^n), n=1..infinity). - 
               Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 15 2004
%F A002654 Expansion of (eta(q^2)^10/(eta(q)eta(q^4))^4 -1)/4 in powers of q.
%F A002654 G.f.: Sum_{k>0} x^k/(1+x^(2k)) = Sum_{k>0} -(-1)^k x^(2k-1)/(1-x^(2k-1)). 
               - Michael Somos Aug 17 2005
%F A002654 a(4n+3)=a(9n+3)=a(9n+6)=0. a(9n)=a(2n)=a(n). - Michael Somos Nov 01 2006
%F A002654 a(n) = SUM(A010052(k)*A010052(n-k): 1<=k<=n). [From Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Sep 27 2008]
%e A002654 4=2^2, so a(4)=1; 5=1^2+2^2=2^2+1^2, so a(5)=2.
%p A002654 with(numtheory): A002654 := proc(n) local it, count1, count3, i; it := 
               divisors(n): count1 := 0: count3 := 0: for i from 1 to tau(n) do 
               if it[i] mod 4 = 1 then count1 := count1+1 fi: if it[i] mod 4 = 3 
               then count3 := count3+1 fi: count1-count3; end;
%o A002654 (PARI) direuler(p=2,101,1/(1-(kronecker(-4,p)*(X-X^2))-X))
%o A002654 (PARI) a(n)=if(n<1,0,polcoeff(sum(k=1,n,x^k/(1+x^(2*k)),x*O(x^n)),n))
%o A002654 (PARI) a(n)=if(n<1,0,sumdiv(n,d,(d%4==1)-(d%4==3)))
%o A002654 (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^10/
               (eta(x+A)*eta(x^4+A))^4/4, n))} /* Michael Somos Jun 03 2005 */
%Y A002654 Cf. A000161, A001481. Equals 1/4 of A004018. Partial sums give A014200.
%Y A002654 A008441(n)=a(4n+1). A122856(n)=a(3n+2). A122865(n)=a(3n+1). A002175(n)=a(12n+1). 
               A121444(n)=a(12n+5)/2.
%Y A002654 Cf. A022544, A143574. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 27 2008]
%Y A002654 Sequence in context: A124764 A151899 A079632 this_sequence A113652 A106139 
               A052154
%Y A002654 Adjacent sequences: A002651 A002652 A002653 this_sequence A002655 A002656 
               A002657
%K A002654 core,easy,nonn,nice,mult
%O A002654 1,5
%A A002654 N. J. A. Sloane (njas(AT)research.att.com).
%E A002654 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000

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