login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
37
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

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(A022544(n)) = 0; a(A001481(n)) > 0. - Reinhard Zumkeller, Sep 27 2008

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.

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)

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.

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. [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. [Annotated scanned copy of pages 104-107 only]

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

Index entries for sequences related to sums of squares

Index entries for sequences related to sublattices

Index entries for "core" sequences

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(A010052(k)*A010052(n-k): 1<=k<=n). - 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

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[105] (* 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, A213408. 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). A008441.

Sequence in context: A209314 A079632 * A113652 A106139 A052154 A039977

Adjacent sequences:  A002651 A002652 A002653 * A002655 A002656 A002657

KEYWORD

core,easy,nonn,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 21 07:08 EDT 2017. Contains 290862 sequences.