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Search: id:A000143
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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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6
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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
(list; graph; listen)
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OFFSET
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0,2
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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.
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. 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
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.
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)=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
G.f.: 1 +16 Sum_{k>0} k^3 x^k/(1-(-x)^k) . - Michael Somos Sep 25 2005
Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos Sep 25 2005
Expansion of (eta(q^2)^5/(eta(q)eta(q^4))^2)^8 in powers of q. - Michael Somos Sep 25 2005
Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos Mar 21 2008
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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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Needs["NumberTheory`NumberTheoryFunctions`"]; Table[SumOfSquaresR[8, n], {n, 0, 33}] (*Chandler*)
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PROGRAM
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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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A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.
Adjacent sequences: A000140 A000141 A000142 this_sequence A000144 A000145 A000146
Sequence in context: A056001 A053526 A107908 this_sequence A035016 A081194 A121148
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KEYWORD
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nonn,easy
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AUTHOR
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njas
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