A053604 Number of ways to write n as an ordered sum of 3 nonzero triangular numbers.

0, 0, 0, 1, 0, 3, 0, 3, 3, 1, 6, 0, 6, 3, 6, 3, 3, 9, 1, 12, 0, 6, 9, 6, 6, 6, 9, 6, 12, 0, 10, 9, 12, 6, 9, 9, 3, 18, 3, 12, 12, 9, 9, 9, 12, 10, 12, 9, 9, 18, 6, 6, 27, 6, 12, 6, 9, 18, 15, 15, 6, 21, 9, 13, 12, 9, 18, 21, 9, 6, 21, 15, 15, 15, 12, 15, 18, 15, 9

Offset: 0

Author

N. J. A. Sloane, Jan 20 2000

Keywords

• nonn

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.

References

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

Links

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

Crossrefs

Cf. A000217, A007294, A051611, A051533, A053603, A008443, A002636.

Programs

Mathematica

nmax = 100; m0 = 10; A053604 :=
Table[a[n], {n, 0, nmax}]; Clear[counts];
counts[m_] :=
counts[m] = (Clear[a]; a[_] = 0;
   Do[s = i*(i + 1)/2 + j*(j + 1)/2 + k*(k + 1)/2;
    a[s] = a[s] + 1, {i, 1, m}, {j, 1, m}, {k, 1, m}];
   A053603); counts[m = m0]; counts[m = 2*m]; While[
counts[m] != counts[m/2], m = 2*m]; A053604  (* G. C. Greubel, Dec 24 2016 *)

Formula

G.f.: ( Sum_{k>=1} x^(k*(k+1)/2) )^3. - Ilya Gutkovskiy, Dec 24 2016