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

0,6

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

Formula

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

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 *)

Crossrefs

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

Sequence in context: A096597 A097994 A318050 * A066958 A357062 A066851

Adjacent sequences: A053601 A053602 A053603 * A053605 A053606 A053607

Keyword

nonn

Author

N. J. A. Sloane, Jan 20 2000

Status

approved