1,5

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.

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

a(A022544(n)) = 0; a(A001481(n)) > 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 27 2008]

M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

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)

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 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.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15.

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, 156163. [See Table 1]. [From N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]

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).

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Baake and U. Grimm, Quasicrystalline combinatorics

Index entries for sequences related to sums of squares

Index entries for sequences related to sublattices

Index entries for "core" sequences

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 ].

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.: 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

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^(2k)) = Sum_{k>0} -(-1)^k x^(2k-1)/(1-x^(2k-1)). - Michael Somos Aug 17 2005

a(4n+3)=a(9n+3)=a(9n+6)=0. a(9n)=a(2n)=a(n). - Michael Somos Nov 01 2006

a(n) = SUM(A010052(k)*A010052(n-k): 1<=k<=n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 27 2008]

4=2^2, so a(4)=1; 5=1^2+2^2=2^2+1^2, so a(5)=2.

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;

(PARI) direuler(p=2, 101, 1/(1-(kronecker(-4, p)*(X-X^2))-X))

(PARI) a(n)=if(n<1, 0, polcoeff(sum(k=1, n, x^k/(1+x^(2*k)), x*O(x^n)), n))

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, (d%4==1)-(d%4==3)))

(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 */

Cf. A000161, A001481. Equals 1/4 of A004018. Partial sums give A014200.

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.

Cf. A022544, A143574. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 27 2008]

Sequence in context: A124764 A151899 A079632 this_sequence A113652 A106139 A052154

Adjacent sequences: A002651 A002652 A002653 this_sequence A002655 A002656 A002657

core,easy,nonn,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000