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

1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 28, 33, 40, 46, 56, 64, 75, 84, 97, 109, 125, 137, 156, 170, 192, 210, 232, 251, 276, 296, 322, 347, 376, 400, 435, 463, 498, 529, 567, 600, 641, 674, 720, 758, 808, 849, 901, 942, 1001, 1051, 1110, 1157, 1225, 1275

Offset: 0

Author

Robert Price, Mar 02 2013

Keywords

• nonn
• easy

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Robert Price)
• Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.

Crossrefs

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

Programs

Maple

h:= proc() true end:
b:= proc(n) local c, i, j, p;
      c:=0;
      for i to iquo(n, 3) do
        for j from i to iquo(n-i, 2) do p:= i*j*(n-i-j);
          if h(p) then h(p):= false; c:=c+1 fi
        od
      od; c
    end:
a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 02 2013

Mathematica

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}]

Formula

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