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A008443
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Number of ordered ways of writing n as the sum of 3 triangular numbers.
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52
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1, 3, 3, 4, 6, 3, 6, 9, 3, 7, 9, 6, 9, 9, 6, 6, 15, 9, 7, 12, 3, 15, 15, 6, 12, 12, 9, 12, 15, 6, 13, 21, 12, 6, 15, 9, 12, 24, 9, 18, 12, 9, 18, 15, 12, 13, 24, 9, 15, 24, 6, 18, 27, 6, 12, 15, 18, 24, 21, 15, 12, 27, 9, 13, 18, 15, 27, 27, 9, 12, 27, 15, 24, 21, 12, 15, 30, 15, 12
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OFFSET
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0,2
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COMMENTS
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Fermat asserted that every number is the sum of three triangular numbers. This was proved by Gauss, who recorded in his Tagebuch entry for Jul 10 1796 that: EYPHEKA! num = DELTA + DELTA + DELTA. See also Gauss, DA, art. 293.
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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. 102.
C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, 1966, New Haven and London, p. 342, art. 293.
M. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.
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LINKS
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George E. Andrews, EYPHKA! num = Delta + Delta + Delta, J. Number Theory 23 (1986), 285-293. [The Y in the title is really the Greek letter Upsilon and Delta is really the Greek letter of that name.]
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FORMULA
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Expansion of Jacobi theta constant theta_2^3 /8. G.f. is cube of g.f. for A010054.
Expansion of psi(q)^3 in powers of q where psi() is a Ramanujan theta function (A010054). - Michael Somos, Oct 25 2006
Expansion of q^(-3/8) * (eta(q^2)^2 / eta(q))^3 in powers of q. - Michael Somos, May 29 2012
Euler transform of period 2 sequence [ 3, -3, ...]. - Michael Somos, Oct 25 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(-3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A213384. - Michael Somos, Jun 23 2012
G.f.: (Sum_{k>0} x^((k^2 - k)/2))^3 = (Product_{k>0} (1 + x^k) * (1 - x^(2*k)))^3. - Michael Somos, May 29 2012
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EXAMPLE
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5 can be written as 3+1+1, 1+3+1, 1+1+3, so a(5) = 3.
G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 6*x^4 + 3*x^5 + 6*x^6 + 9*x^7 + 3*x^8 + ...
G.f. = q^3 + 3*q^11 + 3*q^19 + 4*q^27 + 6*q^35 + 3*q^43 + 6*q^51 + 9*q^59 + 3*q^67 + ...
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MAPLE
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s1 := sum(q^(n*(n+1)/2), n=0..30): s2 := series(s1^3, q, 250): for i from 0 to 200 do printf(`%d, `, coeff(s2, q, i)) od:
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MATHEMATICA
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s1 = Sum[q^(n (n + 1)/2), {n, 0, 12}]; s2 = Series[s1^3, {q, 0, 80}]; CoefficientList[s2, q] (* Jean-François Alcover, Oct 04 2011, after Maple *)
a[ n_] := SeriesCoefficient[ (1/8) EllipticTheta[ 2, 0, q]^3, {q, 0, 2 n + 3/4}]; (* Michael Somos, May 29 2012 *)
QP = QPochhammer; CoefficientList[(QP[q^2]^2/QP[q])^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^((k^2 + k)/2), 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^2 + A)^2 / eta(x + A))^3, n))}; /* Michael Somos, Oct 25 2006 */
(Magma) Basis( ModularForms( Gamma0(16), 3/2), 630)[4] /* Michael Somos, Aug 26 2015 */
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CROSSREFS
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Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440,A226252, A007331, A226253, A226254, A226255, A014787, A014809.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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