A000143 - OEIS

%I #106 Jul 12 2024 06:21:00

%S 1,16,112,448,1136,2016,3136,5504,9328,12112,14112,21312,31808,35168,

%T 38528,56448,74864,78624,84784,109760,143136,154112,149184,194688,

%U 261184,252016,246176,327040,390784,390240,395136,476672,599152,596736,550368,693504,859952

%N Number of ways of writing n as a sum of 8 squares.

%C The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - _Wolfdieter Lang_, Dec 08 2016

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).

%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 314 - 315.

%H T. D. Noe, <a href="/A000143/b000143.txt">Table of n, a(n) for n = 0..10000</a>

%H J. M. Borwein and K.-K. S. Choi, <a href="http://dx.doi.org/10.1023/A:1026230709945">On Dirichlet series for sums of squares</a>, Ramanujan J. 7 (1-3) (2003) 95-127.

%H H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.

%H Shi-Chao Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2010.01.011">Congruences for rs(n)</a>, Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.

%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

%H P. J. C. Lamont, <a href="https://doi.org/10.1017/S001309150000420X">The number of Cayley integers of given norm</a>, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).

%H S. C. Milne, <a href="http://dx.doi.org/10.1023/A:1014865816981">Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions</a>, Ramanujan J., 6 (2002), 7-149.

%H M. Peters, <a href="http://dx.doi.org/10.4064/aa102-2-2">Sums of nine squares</a>, Acta Arith., 102 (2002), 131-135.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0<d|n} (-1)^d*d^3.

%F Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Mar 21 2008

%F Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - _Michael Somos_, Sep 25 2005

%F G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

%F Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - _Michael Somos_, Apr 10 2005

%F a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - _Michael Somos_, Sep 25 2005

%F G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - _Michael Somos_, Sep 25 2005

%F A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

%F Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], _R. J. Mathar_, Jul 02 2012

%F a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017

%F Sum_{k=1..n} a(k) ~ Pi^4 * n^4 /24. - _Vaclav Kotesovec_, Jul 12 2024

%e 1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

%p (sum(x^(m^2),m=-10..10))^8;

%p with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n),n=0..100)]; # _N. J. A. Sloane_, Sep 15 2018

%t Table[SquaresR[8, n], {n, 0, 33}] (* _Ray Chandler_, Dec 06 2006 *)

%t SquaresR[8,Range[0,50]] (* _Harvey P. Dale_, Aug 26 2011 *)

%t QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 01 2015, adapted from PARI *)

%o (PARI) {a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* _Michael Somos_, Sep 25 2005 */

%o (SageMath)

%o Q = DiagonalQuadraticForm(ZZ, [1]*8)

%o Q.representation_number_list(60) # _Peter Luschny_, Jun 20 2014

%o (Julia) # JacobiTheta3 is defined in A000122.

%o A000143List(len) = JacobiTheta3(len, 8)

%o A000143List(37) |> println # _Peter Luschny_, Mar 12 2018

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A000143(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items())<<4 if n else 1 # _Chai Wah Wu_, Jun 21 2024

%Y 8th column of A286815. - _Seiichi Manyama_, May 27 2017

%Y Row d=8 of A122141.

%Y Cf. A008457, A035016, A010815.

%Y Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%Y Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_