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 A053604 Number of ways to write n as an ordered sum of 3 nonzero triangular numbers. 15
 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 (list; graph; refs; listen; history; text; internal format)
 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, A053604, A053603, A008443, A002636. Sequence in context: A096597 A097994 A318050 * A066958 A066851 A288571 Adjacent sequences:  A053601 A053602 A053603 * A053605 A053606 A053607 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 20 2000 STATUS approved

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Last modified July 26 08:42 EDT 2021. Contains 346294 sequences. (Running on oeis4.)