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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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17
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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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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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.
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REFERENCES
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Andrews, George E., 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.]
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
M. Doring, J. Haidenbauer, U.-G. Meissner and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, Arxiv preprint arXiv:1108.0676, 2011.
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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T. D. Noe, Table of n, a(n) for n=0..5050
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. D. Hirschhorn & J. A. Sellers, Partitions Into Three Triangular Numbers
M. D. Hirschhorn & J. A. Sellers, On Representations Of A Number As A Sum Of Three Triangles
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FORMULA
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Expansion of Jacobi theta constant theta_2^3 /8.
Expansion of psi(q)^3 in powers of q where psi() is a Ramanujan theta functions. - 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 g.f. for A213384. - Michael Somos, Jun 23 2012
a(3*n) = A213627(n). a(3*n + 1) = 3 * A213617(n). a(3*n + 2) = A181648(n). - 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.
1 + 3*x + 3*x^2 + 4*x^3 + 6*x^4 + 3*x^5 + 6*x^6 + 9*x^7 + 3*x^8 + ...
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] (* From 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 *)
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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) = local(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 */
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CROSSREFS
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Cf. A053604, A002636.
Partial sums are in A038835. a(n) = A005869(n)/2 = A005886(n)/4 = A005878(n)/8.
Sequence in context: A043551 A162888 A151759 * A196456 A196485 A196718
Adjacent sequences: A008440 A008441 A008442 * A008444 A008445 A008446
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, Feb 07 2001
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STATUS
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approved
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