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A213213 Number of distinct products i*j*k over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.

%I A213213 #20 Sep 23 2024 11:57:56
%S A213213 1,1,1,2,3,5,7,10,13,18,22,28,33,40,46,56,64,75,84,97,109,125,137,156,
%T A213213 170,192,210,232,251,276,296,322,347,376,400,435,463,498,529,567,600,
%U A213213 641,674,720,758,808,849,901,942,1001,1051,1110,1157,1225,1275
%N A213213 Number of distinct products i*j*k over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.
%C A213213 This sequence is in reply to an extension request made in A100450.
%H A213213 Alois P. Heinz, <a href="/A213213/b213213.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..200 from Robert Price)
%H A213213 Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, <a href="https://arxiv.org/abs/2409.11268">Elementary symmetric partitions</a>, arXiv:2409.11268 [math.CO], 2024. See p. 20.
%F A213213 a(n) = (A213207(n)+1)/2.
%p A213213 h:= proc() true end:
%p A213213 b:= proc(n) local c, i, j, p;
%p A213213       c:=0;
%p A213213       for i to iquo(n, 3) do
%p A213213         for j from i to iquo(n-i, 2) do p:= i*j*(n-i-j);
%p A213213           if h(p) then h(p):= false; c:=c+1 fi
%p A213213         od
%p A213213       od; c
%p A213213     end:
%p A213213 a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:
%p A213213 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 02 2013
%t A213213 f[n_] := Length[ Union[ Flatten[ Table[ If[ i+j+k <= n, i*j*k, 0], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]]]; Table[ f[n], {n, 0, 200}]
%Y A213213 Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212.
%K A213213 nonn,easy
%O A213213 0,4
%A A213213 _Robert Price_, Mar 02 2013