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%I #94 Dec 18 2023 12:08:09
%S 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,
%T 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,
%U 15,18,24,21,15,12,27,9,13,18,15,27,27,9,12,27,15,24,21,12,15,30,15,12
%N Number of ordered ways of writing n as the sum of 3 triangular numbers.
%C 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.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Andrews (2016), Theorem 2, shows that A008443(n) = A290735(n) + A290737(n) + A290739(n). = _N. J. A. Sloane_, Aug 10 2017
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%D C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, 1966, New Haven and London, p. 342, art. 293.
%D M. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.
%H N. J. A. Sloane, <a href="/A008443/b008443.txt">Table of n, a(n) for n = 0..20000</a> (first 5050 terms from T. D. Noe)
%H George E. Andrews, <a href="http://dx.doi.org/10.1016/0022-314X(86)90074-0">EYPHKA! num = Delta + Delta + Delta</a>, 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.]
%H George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016.
%H M. Doring, J. Haidenbauer, U.-G. Meissner, and A. Rusetsky, <a href="http://arxiv.org/abs/1108.0676">Dynamical coupled-channel approaches on a momentum lattice</a>, arXiv preprint arXiv:1108.0676 [hep-lat], 2011.
%H M. D. Hirschhorn and J. A. Sellers, <a href="https://ajc.maths.uq.edu.au/pdf/30/ajc_v30_p307.pdf">Partitions into three triangular numbers</a>, Australasian Journal of Combinatorics, Volume 30 (2004), Pages 307-318; <a href="http://www.math.psu.edu/sellersj/3tri_pars_australasian_submission.pdf">Submission</a>.
%H M. D. Hirschhorn and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/p7.pdf">On Representations Of A Number As A Sum Of Three Triangles</a>, Acta Arithmetica 77 (1996), 289-301.
%H K. Ono, S. Robins, and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of Jacobi theta constant theta_2^3 /8. G.f. is cube of g.f. for A010054.
%F Expansion of psi(q)^3 in powers of q where psi() is a Ramanujan theta function (A010054). - _Michael Somos_, Oct 25 2006
%F Expansion of q^(-3/8) * (eta(q^2)^2 / eta(q))^3 in powers of q. - _Michael Somos_, May 29 2012
%F Euler transform of period 2 sequence [ 3, -3, ...]. - _Michael Somos_, Oct 25 2006
%F 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
%F a(3*n) = A213627(n). a(3*n + 1) = 3 * A213617(n). a(3*n + 2) = A181648(n). - _Michael Somos_, Jun 23 2012
%F 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
%F a(n) = A005869(n)/2 = A005886(n)/4 = A005878(n)/8.
%F a(n) = A005875(8*n+3)/8. See, e.g., the Ono et al. link: The case k=3. - _Wolfdieter Lang_, Jan 12 2017
%F a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 06 2017
%e 5 can be written as 3+1+1, 1+3+1, 1+1+3, so a(5) = 3.
%e 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 + ...
%e 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 + ...
%p 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:
%t 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 *)
%t a[ n_] := SeriesCoefficient[ (1/8) EllipticTheta[ 2, 0, q]^3, {q, 0, 2 n + 3/4}]; (* _Michael Somos_, May 29 2012 *)
%t QP = QPochhammer; CoefficientList[(QP[q^2]^2/QP[q])^3 + O[q]^80, q] (* _Jean-François Alcover_, Nov 24 2015 *)
%o (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 */
%o (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 */
%o (Magma) Basis( ModularForms( Gamma0(16), 3/2), 630)[4] /* _Michael Somos_, Aug 26 2015 */
%Y 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.
%Y Cf. A053604, A002636.
%Y Partial sums are in A038835.
%Y Cf. A005869, A005875, A005878, A005886.
%Y Cf. A290733-A290740.
%K nonn,easy,look,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Feb 07 2001
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