%I #20 Sep 23 2024 11:57:56

%S 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 170,192,210,232,251,276,296,322,347,376,400,435,463,498,529,567,600,

%U 641,674,720,758,808,849,901,942,1001,1051,1110,1157,1225,1275

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

%C This sequence is in reply to an extension request made in A100450.

%H 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 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 a(n) = (A213207(n)+1)/2.

%p h:= proc() true end:

%p b:= proc(n) local c, i, j, p;

%p c:=0;

%p for i to iquo(n, 3) do

%p for j from i to iquo(n-i, 2) do p:= i*j*(n-i-j);

%p if h(p) then h(p):= false; c:=c+1 fi

%p od

%p od; c

%p end:

%p a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Mar 02 2013

%t 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 Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212.

%K nonn,easy

%O 0,4

%A _Robert Price_, Mar 02 2013