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A004015
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Theta series of face-centered cubic (f.c.c.) lattice.
(Formerly M4821)
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18
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1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24, 48, 24, 48, 8, 84, 24, 96, 48, 24, 0, 96, 6, 96, 48, 48, 36, 120, 24, 48, 24, 48, 48, 120, 24, 120, 0, 96, 24, 108, 30, 48, 72, 72, 32, 144, 0, 96, 72, 72, 48, 120, 0, 144, 12, 48, 48, 168, 48, 96
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 113.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 263.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.
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LINKS
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FORMULA
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Expansion of phi(q^2)^3 + 12 * q * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
Expansion of (phi(q)^3 + phi(-q)^3) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of b(q) * phi(q^18) + c(q^3) * phi(q^2) in powers of q^3 where b(), c() are cubic AGM theta functions and phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004013.
G.f.: Sum_{i, j, k in Z} x^( i*i + j*j + k*k + i*j + i*k + j*k ). - Michael Somos, Jan 02 2012
Number of integer solutions to x^2 + y^2 + z^2 + x*y + x*z + y*z = n.
Number of integer solutions to x + y + z even and x^2 + y^2 + z^2 = 2 * n.
Number of integer solutions to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 2 * n. (End)
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EXAMPLE
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G.f. = 1 + 12*x + 6*x^2 + 24*x^3 + 12*x^4 + 24*x^5 + 8*x^6 + 48*x^7 + 6*x^8 + ...
G.f. = 1 + 12*q^2 + 6*q^4 + 24*q^6 + 12*q^8 + 24*q^10 + 8*q^12 + 48*q^14 + 6*q^16 + ...
a(2) = 6 since (1, -1, -1) is a solution to x^2 + y^2 + z^2 + x*y + x*z + y*z = 2 and the other 5 solutions are permutations and negations of this one.
a(2) = 6 since (1, 1, -1, -1) is a solution to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 4 and the other 5 solutions are permutations of this one. (End)
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MAPLE
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maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a, q, maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a, q, maxd); th4 := series(subs(q=-q, th3), q, maxd); series((1/2)*(th3^3+th4^3), q, 200);
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 + EllipticTheta[ 4, 0, q]^3) / 2, {q, 0, 2 n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^3 + 12 q QPochhammer[ q^4]^3 QPochhammer[ q^8]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n*=2; polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, Oct 25 2006 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / eta(x^2 + A)^2 / eta(x^8 + A)^2)^3 + 12 * x * eta(x^4 + A)^3 * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))}; /* Michael Somos, May 17 2008 */
(PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 1; 1, 2, 1; 1, 1, 2], n, 1)[n])}; /* Michael Somos, Jan 02 2012 */
(Magma) L := Lattice("A", 3); A<q> := ThetaSeries(L, 140); A; /* Michael Somos, Nov 13 2014 */
(Magma) A := Basis( ModularForms( Gamma1(8), 3/2), 70); A[1] + 12*A[2] + 6*A[3] + 24*A[4]; /* Michael Somos, Sep 08 2018 */
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CROSSREFS
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Cf. A000007, A000122, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_4, ...)
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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