A000118 Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Number 5 and 35 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

This is the convolution square of A004018. - Pierre Abbat, May 15 2023

Examples

			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 + ...
a(1)=8 counts 1 = 1^2 + 0^2 + 0^2 + 0^2 = 0^2 + 1^2 + 0^2 + 0^2 = 0^2 + 0^2 + 1^2 + 0^2 = 0^2 + 0^2 + 0^2 + 1^2 and 4 more sums where 1^2 is replaced by (-1)^2. - _R. J. Mathar_, May 16 2023
		

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.
• 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

• N. J. A. Sloane, Table of n, a(n) for n = 0..50000 (first 10000 terms from T. D. Noe)
• George E. Andrews, S. B. Ekhad, and 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.
• George E. Andrews, S. B. Ekhad, and D. Zeilberger, A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a sum of Four Squares
• George E. Andrews, Sumit Kumar Jha, and J. López-Bonilla, Sums of Squares, Triangular Numbers, and Divisor Sums, Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.5.
• Michael Ball and Dario Alejandro Alpern, Every positive integer is a sum of four integer squares
• Cristina Ballantine and Mircea Merca, Jacobi's four and eight squares theorems and partitions into distinct parts, Mediterr. J. Math 16 (2009) 26.
• 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).
• Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
• 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
• Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
• G. Nebe and N. J. A. Sloane, The lattice Z4
• 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
• Min Wang and 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, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
• 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

Crossrefs

Row d=4 of A122141 and of A319574, 4th column of A286815.

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.

Programs

Haskell

a000118 0 = 1
a000118 n = 8 * a046897 n  -- Reinhard Zumkeller, Aug 12 2015

Julia

# JacobiTheta3 is defined in A000122.
A000118List(len) = JacobiTheta3(len, 4)
A000118List(57) |> println # Peter Luschny, Mar 12 2018

MATLAB

a(n) = 8 * sum(find(mod(n,1:n)==0 & mod(1:n,4))) + (n==0) % David Mellinger, Aug 04 2025

Magma

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

Maple

(add(q^(m^2),m=-10..10))^4; seq(coeff(%,q,n), n=0..50);
# Alternative:
A000118list := proc(len) series(JacobiTheta3(0, x)^4, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000118list(57); # Peter Luschny, Oct 02 2018

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-François Alcover, Nov 24 2015 *)

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

Python

from sympy import divisors
def a(n): return 1 if n==0 else 8*sum(d for d in divisors(n) if d%4 != 0)
print([a(n) for n in range(57)]) # Michael S. Branicky, Jan 08 2021

Python

from sympy import divisor_sigma
def A000118(n): return 1 if n == 0 else 8*divisor_sigma(n) if n % 2 else 24*divisor_sigma(int(bin(n)[2:].rstrip('0'),2)) # Chai Wah Wu, Jun 27 2022

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

Formula

G.f.: theta_3(q)^4 = (Product_{n>=1} (1-q^(2n))*(1+q^(2n-1))^2)^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.

a(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)

a(n) = (8/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017