A008443 Number of ordered ways of writing n as the sum of 3 triangular numbers.

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

Offset

0,2

Comments

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.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Andrews (2016), Theorem 2, shows that A008443(n) = A290735(n) + A290737(n) + A290739(n). = N. J. A. Sloane, Aug 10 2017

References

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.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (First 5050 terms from T. D. Noe)

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.]

George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016.

M. Doring, J. Haidenbauer, U.-G. Meissner and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv preprint arXiv:1108.0676 [hep-lat], 2011.

M. D. Hirschhorn & J. A. Sellers, Partitions into three triangular numbers, Australasian Journal of Combinatorics, Volume 30 (2004), Pages 307-318; Submission.

M. D. Hirschhorn & J. A. Sellers, On Representations Of A Number As A Sum Of Three Triangles, Acta Arithmetica 77 (1996), 289 - 301.

K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

a(n) = A005869(n)/2 = A005886(n)/4 = A005878(n)/8.

a(n) = A005875(8*n+3)/8. See, e.g., the Ono et al. link: The case k=3. - Wolfdieter Lang, Jan 12 2017

a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

Example

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 + ...

Maple

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:

Mathematica

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 *)

Prog

(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 */

Crossrefs

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.

Cf. A053604, A002636.

Partial sums are in A038835.

Cf. A005869, A005875, A005878, A005886.

Cf. A290733-A290740.

Sequence in context: A162888 A337402 A151759 * A196456 A196485 A196718

Adjacent sequences: A008440 A008441 A008442 * A008444 A008445 A008446

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 07 2001

Status

approved