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%I #12 Feb 18 2019 16:42:50
%S 1,1,1,2,1,2,2,2,2,3,2,3,2,3,3,2,4,3,4,2,3,4,4,2,5,4,2,5,3,6,2,3,5,6,
%T 3,4,5,5,4,3,6,4,5,5,5,5,5,2,7,6,5,5,3,6,6,4,6,8,3,6,5,7,5,3,8,6,6,5,
%U 6,8,5,4,8,6,4,7,7,6,7,2,8,10,6,6,5,7,6
%N Number of ways to write n as floor(i^2/3) + floor(j^2/3) + floor(k^2/3) with 1 <= i <= j <= k.
%C Farhi proved that a(n) > 0 for any n >= 0.
%C i^2/3 means (i^2)/3, of course, not i^(2/3). - _N. J. A. Sloane_, Feb 18 2019
%H Rémy Sigrist, <a href="/A306468/b306468.txt">Table of n, a(n) for n = 0..10000</a>
%H Bakir Farhi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Farhi/farhi12.html">An Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence floor(n^2/3)</a>, Journal of Integer Sequences, Vol. 17 (2014), #14.7.6.
%H Rémy Sigrist, <a href="/A306468/a306468.gp.txt">PARI program for A306468</a>
%e For n = 42:
%e - let f(k) = floor(k^2/3),
%e - 42 can be written in 5 ways as f(i) + f(j) + f(k) with 1 <= i <= j <= k:
%e f(1) + f(8) + f(8) = 0 + 21 + 21
%e f(2) + f(2) + f(11) = 1 + 1 + 40
%e f(2) + f(5) + f(10) = 1 + 8 + 33
%e f(3) + f(6) + f(9) = 3 + 12 + 27
%e f(4) + f(7) + f(8) = 5 + 16 + 21,
%e - hence a(42) = 5.
%o (PARI) See Links section.
%Y Cf. A000212.
%K nonn
%O 0,4
%A _Rémy Sigrist_, Feb 17 2019