0,2

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.

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. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.

T. D. Noe, Table of n, a(n) for n=0..5050

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

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

Euler transform of period 2 sequence [ 3, -3, ...]. - Michael Somos Oct 25 2006

5 can be written as 3+1+1, 1+3+1, 1+1+3, so a(5) = 3.

1 + 3*q + 3*q^2 + 4*q^3 + 6*q^4 + 3*q^5 + 6*q^6 + 9*q^7 + 3*q^8 + ... - Michael Somos Aug 17 2009

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:

(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 */

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 this_sequence A074883 A092200 A130481

Adjacent sequences: A008440 A008441 A008442 this_sequence A008444 A008445 A008446

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 07 2001