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 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). (Formerly M0012 N0001) 53
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Glaisher calls this E(n) or E_0(n). - N. J. A. Sloane, Nov 24 2018 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. a(n) is also one fourth of the number of integer solutions of n = x^2 + y^2 (order and signs matter, and 0 (without signs) is allowed). a(n) = N(n)/4, with N(n) from p. 147 of the Niven-Zuckermann reference. See also Theorem 5.12, p. 150, which defines a (strongly) multiplicative function h(n) which coincides with A056594(n-1), n >= 1, and N(n)/4 = sum(h(d), d divides n). - Wolfdieter Lang, Apr 19 2013 a(2+8*N) = A008441(N) gives the number of ways of writing N as the sum of 2 (nonnegative) triangular numbers for N >= 0. - Wolfdieter Lang, Jan 12 2017 REFERENCES M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44. J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194. 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) E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15. Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), pp. 147 and 150. G. Scheja and U. Storch, Lehrbuch der Algebra, Tuebner, 1988, p. 251. 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 T. D. Noe, Table of n, a(n) for n = 1..10000 M. Baake and U. Grimm, Quasicrystalline combinatorics 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 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. 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. [Annotated scanned copy of pages 104-107 only] 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). 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. John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From N. J. A. Sloane, Feb 23 2009 FORMULA 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 zeta-function of Z[ i ]. Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -16. 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). 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 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) * (4*w + 1). - Michael Somos, Jul 19 2004 G.f.: Sum((-1)^floor(n/2)*x^((n^2+n)/2)/(1+(-x)^n), n=1..infinity). - Vladeta Jovovic, Sep 15 2004 Expansion of (eta(q^2)^10 / (eta(q) * eta(q^4))^4 - 1)/4 in powers of q. G.f.: Sum_{k>0} x^k / (1 + x^(2*k)) = Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 - x^(2*k - 1)). - Michael Somos, Aug 17 2005 a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(2*n) = a(n). - Michael Somos, Nov 01 2006 a(4*n + 1) = A008441(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n). 4 * a(n) = A004018(n) unless n=0. a(n) = Sum_{k=1..n} A010052(k)*A010052(n-k). a(A022544(n)) = 0; a(A001481(n)) > 0. - Reinhard Zumkeller, Sep 27 2008 a(n) = A001826(n) - A001842(n). - R. J. Mathar, Mar 23 2011 a(n) = sum(A056594(d-1), d divides n), n >= 1. See the above comment on A056594(d-1) = h(d) of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013 Dirichlet g.f.: zeta(s)*beta(s) = zeta(s)*L(chi_2(4),s). - Ralf Stephan, Mar 27 2015 G.f.: (theta_3(x)^2 - 1)/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018 a(n) = Sum_{ m: m^2|n } A000089(n/m^2). - Andrey Zabolotskiy, May 07 2018 a(n) = A053866(n) + 2 * A025441(n). - Andrey Zabolotskiy, Apr 23 2019 EXAMPLE 4 = 2^2, so a(4) = 1; 5 = 1^2 + 2^2 = 2^2 + 1^2, so a(5) = 2. x + x^2 + x^4 + 2*x^5 + x^8 + x^9 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + x^18 + ... 2 = (+1)^2 + (+1)^2 = (+1)^2 + (-1)^2  = (-1)^2 + (+1)^2 = (-1)^2 + (-1)^2. Hence there are 4 integer solutions, called N(2) in the Niven-Zuckerman reference, and a(2) = N(2)/4 = 1.  4 = 0^1 + (+2)^2 = (+2)^2 + 0^2 = 0^2 + (-2)^2 = (-2)^2 + 0^2. Hence N(4) = 4 and a(4) = N(4)/4 = 1. N(5) = 8, a(5) = 2. - Wolfdieter Lang, Apr 19 2013 MAPLE with(numtheory): A002654 := proc(n)     local count1, count3, d;     count1 := 0:     count3 := 0:     for d in numtheory[divisors](n) do         if d mod 4 = 1 then             count1 := count1+1         elif d mod 4 = 3 then             count3 := count3+1         fi:     end do:     count1-count3; end proc: MATHEMATICA a[n_] := Count[Divisors[n], d_ /; Mod[d, 4] == 1] - Count[Divisors[n], d_ /; Mod[d, 4] == 3]; a/@Range (* Jean-François Alcover, Apr 06 2011, after R. J. Mathar *) QP = QPochhammer; CoefficientList[(1/q)*(QP[q^2]^10/(QP[q]*QP[q^4])^4-1)/4 + O[q]^100, q] (* Jean-François Alcover, Nov 24 2015 *) PROG (PARI) direuler(p=2, 101, 1/(1-X)/(1-kronecker(-4, p)*X)) (PARI) {a(n) = polcoeff( sum(k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)} (PARI) {a(n) = sumdiv( n, d, (d%4==1) - (d%4==3))} (PARI) {a(n) = local(A); 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 */ (PARI) a(n)=my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, (f[i, 2]+1)%2)) \\ Charles R Greathouse IV, Sep 09 2014 (Haskell) a002654 n = product \$ zipWith f (a027748_row m) (a124010_row m) where    f p e | p `mod` 4 == 1 = e + 1          | otherwise      = (e + 1) `mod` 2    m = a000265 n -- Reinhard Zumkeller, Mar 18 2013 CROSSREFS Cf. A000161, A001481. Equals 1/4 of A004018. Partial sums give A014200. Cf. A002175, A008441, A121444, A122856, A122865, A022544, A143574, A000265, A027748, A124010, A025426 (two squares, order does not matter), A120630 (Dirichlet inverse), A101455 (Mobius transform), A000089. If one simply reads the table in Glaisher, PLMS 1884, which omits the zero entries, one gets A213408. Sequence in context: A228347 A209314 A079632 * A113652 A106139 A052154 Adjacent sequences:  A002651 A002652 A002653 * A002655 A002656 A002657 KEYWORD core,easy,nonn,nice,mult AUTHOR STATUS approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)