Search: id:A000143
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%I A000143
%S A000143 1,16,112,448,1136,2016,3136,5504,9328,12112,14112,21312,31808,35168,
%T A000143 38528,56448,74864,78624,84784,109760,143136,154112,149184,194688,
%U A000143 261184,252016,246176,327040,390784,390240,395136,476672,599152,596736
%N A000143 Number of ways of writing n as a sum of 8 squares.
%D A000143 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 77, Eq. (31.61); P. 79 Eq. (32.32).
%D A000143 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, 
               NY, 1985, p. 121.
%D A000143 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 314.
%D A000143 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.
%D A000143 M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
%H A000143 T. D. Noe, <a href="b000143.txt">Table of n, a(n) for n=0..10000</a>
%H A000143 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>
%H A000143 <a href="Sindx_Su.html#ssq">Index entries for sequences related to sums 
               of squares</a>
%F A000143 Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*sum_{0<d|n}(-1)^d*d^3.
%F A000143 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 A000143 Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael 
               Somos, Apr 10 2005
%F A000143 a(n)=16b(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 A000143 G.f.: 1 +16 Sum_{k>0} k^3 x^k/(1-(-x)^k) . - Michael Somos Sep 25 2005
%F A000143 Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael 
               Somos Sep 25 2005
%F A000143 Expansion of (eta(q^2)^5/(eta(q)eta(q^4))^2)^8 in powers of q. - Michael 
               Somos Sep 25 2005
%F A000143 Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta 
               function. - Michael Somos Mar 21 2008
%e A000143 1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 
               + ...
%p A000143 (sum(x^(m^2),m=-10..10))^8;
%t A000143 Needs["NumberTheory`NumberTheoryFunctions`"]; Table[SumOfSquaresR[8, 
               n], {n, 0, 33}] (*Chandler*)
%o A000143 (PARI) a(n)=if(n<1,n==0,16*(-1)^n*sumdiv(n,d,(-1)^d*d^3))
%o A000143 (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 */
%Y A000143 A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.
%Y A000143 Sequence in context: A053526 A107908 A144449 this_sequence A035016 A081194 
               A121148
%Y A000143 Adjacent sequences: A000140 A000141 A000142 this_sequence A000144 A000145 
               A000146
%K A000143 nonn,easy
%O A000143 0,2
%A A000143 N. J. A. Sloane (njas(AT)research.att.com).

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