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

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000118 Number of ways of writing n as a sum of 4 squares; also theta series of lattice Z^4. 132
1, 8, 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112, 192, 192, 24, 144, 312, 160, 144, 256, 288, 192, 96, 248, 336, 320, 192, 240, 576, 256, 24, 384, 432, 384, 312, 304, 480, 448, 144, 336, 768, 352, 288, 624, 576, 384, 96, 456, 744, 576, 336, 432, 960, 576, 192 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a^2+b^2+c^2+d^2 is one of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008

a(n) is also the number of quaternions q = a + bi + cj + dk, where a, b, c, d are integers, such that a^2 + b^2 + c^2 + d^2 = n (i.e., so that n is the norm of q). These are Lipschitz integer quaternions. - Rick L. Shepherd, Mar 27 2009

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, ch. 8, pp. 231-2.

J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, Springer-Verlag, p. 108, Eq. (49).

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.28). See also top of p. 94.

E. Freitag and R. Busam, Funktionentheorie 1, 4. Auflage, Springer, 2006, p. 392.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314, Theorem 386.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of integers, Chapman & Hall/CRC, 2006, p. 29.

S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917) 11-21).

LINKS

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 0..50000 [First 10000 terms from T. D. Noe]

G. E. Andrews, S. B. Ekhad, D. Zeilberger A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares, arXiv:math/9206203 [math.CO], 1992.

G. E. Andrews, S. B. Ekhad, D. Zeilberger, A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a sum of Four Squares

Michael Ball and Dario Alejandro Alpern, Every positive integer is a sum of four integer squares

R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.

R. T. Bumby, Sums of four squares [Cached copy]

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.

Peter L. Clark, A theorem of Minkowski; the four squares theorem (no date).

E. Grosswald, Representations of Integers as Sums of an Even Number of Squares, Springer-Verlag, NY, 1985, p. 121.

M. D. Hirschhorn, A Simple Proof of Jacobi's Four-Square Theorem, Proceedings of the American Mathematical Society, Vol. 101, No. 3 (Nov., 1987), pp. 436-438

G. Nebe and N. J. A. Sloane, Home page for this lattice

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.

Y. Mimura, Almost Universal Quadratic Forms.

Simon Plouffe, Table of n, a(n) for n=0..105817

B. K. Spearman and K. S. Williams, The simplest arithmetic proof of Jacobi's four squares theorem, Far East Journal of Mathematical Sciences 2.3 (2000): 433-440.

Eric van Fossen Conrad, Jacobi's Four Square Theorem [Broken link?]

Min Wang, Zhi-Hong Sun, On the number of representations of n as a linear combination of four triangular numbers II, arXiv:1511.00478 [math.NT], 2015.

Eric W. Weisstein, "Quaternion Norm".

Wikipedia, Hurwitz quaternion

K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

Index entries for sequences related to sums of squares

FORMULA

G.f.: theta_3(q)^4 = Product( (1-q^(2n))*(1+q^(2n-1))^2, n=1..inf )^4 = eta(-q)^8/eta(q^2)^4; eta = Dedekind's function.

a(n) = 8*sigma(n) - 32*sigma(n/4) for n>0, where the latter term is 0 if n is not a multiple of 4.

Euler transform of period 4 sequence [ 8, -12, 8, -4, ...]. - Michael Somos, Dec 16 2002

G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 - 30*u*v^2*w + 12*u*v*w*(u + 9*w) - u*w*(u^2 + 9*w*u + 81*w^2). - Michael Somos, Nov 02 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 4 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 25 2008

For n>0, a(n)/8 is multiplicative and a(p^n)/8 = 1 + p + p^2 + ... + p^n for p an odd prime, a(2^n)/8 = 1 + 2 for n>0.

a(n)=8*A000203(n/A006519(n))*(2+(-1)^n). - Benoit Cloitre, May 16 2002

G.f.: 1 + 8 * Sum_{k>0} x^k / (1 + (-x)^k)^2 = 1 + 8 * Sum_{k>0} k * x^k / (1 + (-x)^k).

G.f. = s(2)^20/(s(1)*s(4))^8, where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

Fine gives another explicit formula for a(n) in terms of the divisors of n.

8*A046897(n), n>0. - Ralf Stephan, Apr 02 2003

A096727(n) = (-1)^n * a(n). a(2*n) = A004011(n). a(2*n + 1) = A005879(n).

Dirichlet g.f. sum_{n>=1} a(n)/n^s = 8*(1-4^(1-s))*zeta(s)*zeta(s-1). [Ramanu. J. 7 (2003) 95-127, eq (3.2)]. - R. J. Mathar, Jul 02 2012

Average value is Pi^2/2 * n + O(sqrt(n)). - Charles R Greathouse IV, Feb 17 2015

From Wolfdieter Lang, Jan 14 2016: (Start)

For n >= 1: a(n) = 8*Sum_{d | n} b(d)*d, with b(d) = 1 if d/4 is not an integer else 0. See, e.g., the Freitag-Busam reference, p. 392.

For n >= 1: a(n) = 8*sigma(n) if n is odd else 24*sigma(m(n)), where m(n) is the largest odd divisor of n (see A000265), and sigma is given in A000203. See the Moreno-Wagstaff reference, Theorem 2. 6 (Jacobi), p. 29. (End)

EXAMPLE

G.f. = 1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + ...

MAPLE

(add(q^(m^2), m=-10..10))^4; seq(coeff(%, q, n), n=0..50);

MATHEMATICA

Table[SquaresR[4, n], {n, 0, 46}]

a[ n_] :=  SeriesCoefficient[ EllipticTheta[ 3, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)

a[ n_] := If[ n < 1, Boole[ n == 0], 8 Sum[ If[ Mod[ d, 4] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Feb 20 2015 *)

QP = QPochhammer; CoefficientList[QP[-q]^8/QP[q^2]^4 + O[q]^60, q] (* Jean-Fran├žois Alcover, Nov 24 2015 *)

PROG

(PARI) {a(n) = if( n<1, n==0, 8 * sumdiv( n, d, if( d%4, d)))}; /* Michael Somos, Apr 01 2003 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2))^4, n))}; /* Michael Somos, Apr 01 2008 */

(PARI) q='q+O('q^66); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4) /* Joerg Arndt, Apr 08 2013 */

(PARI) a(n) = 8*sigma(n) - if (n % 4, 0, 32*sigma(n/4)); \\ Michel Marcus, Jul 13 2016

(Sage) A = ModularForms( Gamma0(4), 2, prec=57) . basis(); A[0] + 8*A[1]; # Michael Somos, Jun 12 2014

(Sage)

Q = DiagonalQuadraticForm(ZZ, [1]*4)

Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

(MAGMA) A := Basis( ModularForms( Gamma0(4), 2), 57); A[1] + 8*A[2]; /* Michael Somos, Aug 21 2014 */

(Haskell)

a000118 0 = 1

a000118 n = 8 * a046897 n  -- Reinhard Zumkeller, Aug 12 2015

CROSSREFS

Cf. A000122, A000203, A000265, A004011, A005879, A046897, A096727.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1, 2, 3, 4, 5, 6, 7, 8, 12, see A000118, A236928, A236926, A236923, A236930,A236931, A236932, A236927, A236933.

Sequence in context: A162829 A175368 * A096727 A028660 A028644 A227175

Adjacent sequences:  A000115 A000116 A000117 * A000119 A000120 A000121

KEYWORD

nonn,easy,nice

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 May 26 16:52 EDT 2017. Contains 287130 sequences.