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A090767 Numbers of the form 3*x*y*z + 2(x*y + y*z + z*x) + (x + y + z) for x, y, z positive integers.

%I A090767 #20 Apr 11 2019 08:55:53
%S A090767 12,20,28,33,36,44,46,52,54,59,60,64,68,72,75,76,82,84,85,92,96,98,
%T A090767 100,104,105,108,111,116,117,118,124,128,132,133,136,137,138,140,144,
%U A090767 148,150,151,154,156,159,162,163,164,170,172,174,176,180,184,188,189,190
%N A090767 Numbers of the form 3*x*y*z + 2(x*y + y*z + z*x) + (x + y + z) for x, y, z positive integers.
%C A090767 This is the set of numbers which count the unit sticks or unit segments needed to construct a three-dimensional cubic lattice made up from unit cubes. This generalizes the two-dimensional version which is A047845 (numbers of the form 2*x*y + x + y for x and y positive integers) and is also the numbers of sticks needed to construct a rectangular lattice of unit squares.
%H A090767 Robert Israel, <a href="/A090767/b090767.txt">Table of n, a(n) for n = 1..10000</a>
%e A090767 a(1) = 12 because there are 12 edges to a cube.
%p A090767 SeqGen1 := proc(n,N) local a,b,c,F,V,v; # n specifies the search space; N specifies the maximal number to appear in the initial segment of the sequence F := 3*x*y*z + 2*(x*y+y*z+z*x)+x+y+z; V := {}; for a from 1 to n do for b from 1 to n do for c from b to n do v := subs(x=a,y=b,F); if v < N then V := V union {v};fi; od;od; sort(V) end:
%p A090767 # alternative:
%p A090767 N:= 1000: # to get all terms <= N
%p A090767 S:= {seq(seq(seq(3*x*y*z + 2*(x*y+y*z+z*x)+(x+y+z),
%p A090767 z = 1 .. min(y, (-2*x*y+N-x-y)/(3*x*y+2*x+2*y+1))),
%p A090767 y = 1 .. min(x, (N-3*x-1)/(5*x+3))),
%p A090767 x = 1 .. (N-4)/8)}:
%p A090767 sort(convert(S,list)); # _Robert Israel_, Feb 18 2016
%t A090767 M = 1000;
%t A090767 S = Table[3 x y z + 2(x y + y z + z x) + (x + y + z), {x, 1, (M - 4)/8}, {y, 1, Min[x, (M - 3 x - 1)/(5 x + 3)]}, {z, 1, Min[y, (-2 x y + M - x - y)/(3 x y + 2 x + 2 y + 1)]}] // Flatten // Union (* _Jean-François Alcover_, Apr 11 2019, after _Robert Israel_ *)
%Y A090767 Cf. A047845.
%K A090767 nonn
%O A090767 1,1
%A A090767 _John H. Mason_, Feb 02 2004
%E A090767 More terms from _Ray Chandler_, Feb 04 2004