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A000118
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Number of ways of writing n as a sum of 4 squares; theta series of lattice Z^4.
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25
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| One of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008
Hence 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, respectively, the square of the norm of q, depending upon apparently varying definitions of "norm"). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 27 2009]
See my previous comment. By definition, any quaternion q = a + bi + cj + dk, where a, b, c, d are integers, is a Lipschitz quaternion (Lipschitz integer), as mentioned today on the seqfan list by Benoit Jubin. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 30 2009]
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REFERENCES
| J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, ch. 8, pp. 231-2. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 27 2009]
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. 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, Theorem 386.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.
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.
S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917) 11-21)
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..10000
D. A. Alpern, Proofs of Lagrange 4 square theorem
G. E. Andrews, S. B. Ekhad, D. Zeilberger [math/9206203] A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares
G. E. Andrews, S. B. Ekhad, D. Zeilberger, A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a sum of Four Squares
R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares
G. Nebe and N. J. A. Sloane, Home page for this lattice
Y. Mimura, Almost Universal Quadratic Forms.
Simon Plouffe, Table of n, a(n) for n=0..105817
E. van Fossen Conrad, Jacobi's Four Square Theorem
Weisstein, Eric W., "Quaternion Norm". [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 27 2009]
Wikipedia, Hurwitz quaternion [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 30 2009]
Index entries for sequences related to sums of squares
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FORMULA
| Euler transform of period 4 sequence [ 8, -12, 8, -4, ...].
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).
G.f. is Fourier series of level 4 weight 2 modular form. 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 (benoit7848c(AT)orange.fr), May 16 2002
G.f.: theta_3(q)^4 = Product( (1-q^(2n))*(1+q^(2n-1))^2, n=1..inf )^4 = eta(-q)^8/eta(q^2)^4; eta = Dedekind's function.
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.
8*A046897(n), n>0. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 02 2003 A096727(n) = (-1)^n * a(n). a(2*n) = A004011(n). a(2*n + 1) = A05879(n).
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EXAMPLE
| 1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + ...
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MAPLE
| (add(q^(m^2), m=-10..10))^4;
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MATHEMATICA
| Table[SquaresR[4, n], {n, 0, 46}]
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PROG
| (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))}
(PARI) {a(n) = if( n<1, n==0, 8 * sumdiv( n, d, if( d%4, d)))}
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CROSSREFS
| Cf. A004011, A05879, A046897, A096727.
Sequence in context: A162829 A175368 * A096727 A028660 A028644 A056196
Adjacent sequences: A000115 A000116 A000117 * A000119 A000120 A000121
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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