Search: id:A000118
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%I A000118
%S A000118 1,8,24,32,24,48,96,64,24,104,144,96,96,112,192,192,24,144,312,160,144,
%T A000118 256,288,192,96,248,336,320,192,240,576,256,24,384,432,384,312,304,480,
%U A000118 448,144,336,768,352,288,624,576,384,96,456,744,576,336,432,960,576,192
%N A000118 Number of ways of writing n as a sum of 4 squares; theta series of lattice 
               Z^4.
%C A000118 One of Ramanujan's 54 universal quaternary quadratic forms. - Michael 
               Somos Apr 01 2008
%C A000118 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]
%C A000118 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]
%D A000118 J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 
               Springer-Verlag, p. 108, Eq. (49).
%D A000118 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 78, Eq. (32.28). See also top of p. 94.
%D A000118 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, 
               NY, 1985, p. 121.
%D A000118 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 314, Theorem 386.
%D A000118 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 A000118 S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 
               (Proceedings of the Camb. Phil. Soc., 19 (1917) 11-21)
%D A000118 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]
%H A000118 T. D. Noe, <a href="b000118.txt">Table of n, a(n) for n=0..10000</a>
%H A000118 D. A. Alpern, <a href="http://www.alpertron.com.ar/4SQUARES.HTM">Proofs 
               of Lagrange 4 square theorem</a>
%H A000118 G. E. Andrews, S. B. Ekhad, D. Zeilberger <a href="http://arXiv.org/abs/
               math.CO/9206203">[math/9206203] A Short Proof of Jacobi's Formula 
               for the Number of Representations of an Integer as a Sum of Four 
               Squares</a>
%H A000118 G. E. Andrews, S. B. Ekhad, D. Zeilberger, <a href="http://www.math.rutgers.edu/
               ~zeilberg/mamarim/mamarimPDF/sum4sq.pdf">A Short Proof of Jacobi's 
               Formula for the Number of Representations of an Integer as a sum 
               of Four Squares</a>
%H A000118 R. T. Bumby, <a href="http://www.math.rutgers.edu/~bumby/squares1.pdf">
               Sums of four squares</a>, in Number theory (New York, 1991-1995), 
               1-8, Springer, New York, 1996.
%H A000118 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 A000118 E. van Fossen Conrad, <a href="http://www.math.ohio-state.edu/~econrad/
               Jacobi/sumofsq/sumofsq.html">Jacobi's Four Square Theorem</a>
%H A000118 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/Z4.html">Home page for this lattice</a>
%H A000118 Simon Plouffe, <a href="a000118.txt">Table of n, a(n) for n=0..105817</
               a>
%H A000118 <a href="Sindx_Su.html#ssq">Index entries for sequences related to sums 
               of squares</a>
%H A000118 Y. Mimura, <a href="http://www.kobepharma-u.ac.jp/~math/notes/note01.html">
               Almost Universal Quadratic Forms</a>.
%H A000118 Weisstein, Eric W., <a href="http://mathworld.wolfram.com/QuaternionNorm.html">
               "Quaternion Norm"</a>. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Mar 27 2009]
%H A000118 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hurwitz_quaternion">
               Hurwitz quaternion</a> [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Jul 30 2009]
%F A000118 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.
%F A000118 a(n)=8*A000203(n/A006519(n))*(2+(-1)^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 16 2002
%F A000118 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.
%F A000118 G.f.: 1+8 Sum_{k>0} x^k/(1+(-x)^k)^2 = 1+8 Sum_{k>0} k*x^k/(1+(-x)^k).
%F A000118 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]
%F A000118 Fine gives another explicit formula for a(n) in terms of the divisors 
               of n.
%F A000118 8*A046897(n), n>0. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 02 
               2003
%F A000118 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).
%F A000118 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
%F A000118 Euler transform of period 4 sequence [ 8, -12, 8, -4, ...].
%e A000118 1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 
               + ...
%p A000118 (add(q^(m^2),m=-10..10))^4;
%t A000118 a[n_] := SumOfSquaresR[4, n]
%o A000118 (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))}
%o A000118 (PARI) {a(n) = if( n<1, n==0, 8 * sumdiv( n, d, if( d%4, d)))}
%Y A000118 A096727(n)=(-1)^n*a(n). A046897(n)=a(n)/8 if n>0. A004011(n)=a(2n). A005879(n)=a(2n+1).
%Y A000118 Sequence in context: A068781 A038524 A162829 this_sequence A096727 A028660 
               A028644
%Y A000118 Adjacent sequences: A000115 A000116 A000117 this_sequence A000119 A000120 
               A000121
%K A000118 nonn,easy,nice
%O A000118 0,2
%A A000118 N. J. A. Sloane (njas(AT)research.att.com).

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