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A000143
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Number of ways of writing n as a sum of 8 squares.
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11
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1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736
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OFFSET
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0,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); P. 79 Eq. (32.32).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..10000
J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.
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.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares
Index entries for sequences related to sums of squares
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FORMULA
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Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*sum_{0<d|n}(-1)^d*d^3.
Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008
Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005
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]
Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005
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
G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005
A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.
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
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EXAMPLE
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1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...
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MAPLE
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(sum(x^(m^2), m=-10..10))^8;
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MATHEMATICA
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Table[SquaresR[8, n], {n, 0, 33}] (* Chandler *)
SquaresR[8, Range[0, 50]] (* From Harvey P. Dale, Aug 26 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}
(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 */
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CROSSREFS
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Cf. A008457, A035016.
Sequence in context: A213754 A144449 * A035016 A205964 A222113 A081194
Adjacent sequences: A000140 A000141 A000142 * A000144 A000145 A000146
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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