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A090767
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Numbers of the form 3*x*y*z + 2(x*y + y*z + z*x) + (x + y + z) for x, y, z positive integers.
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4
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12, 20, 28, 33, 36, 44, 46, 52, 54, 59, 60, 64, 68, 72, 75, 76, 82, 84, 85, 92, 96, 98, 100, 104, 105, 108, 111, 116, 117, 118, 124, 128, 132, 133, 136, 137, 138, 140, 144, 148, 150, 151, 154, 156, 159, 162, 163, 164, 170, 172, 174, 176, 180, 184, 188, 189, 190
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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a(1) = 12 because there are 12 edges to a cube.
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MAPLE
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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:
# alternative:
N:= 1000: # to get all terms <= N
S:= {seq(seq(seq(3*x*y*z + 2*(x*y+y*z+z*x)+(x+y+z),
z = 1 .. min(y, (-2*x*y+N-x-y)/(3*x*y+2*x+2*y+1))),
y = 1 .. min(x, (N-3*x-1)/(5*x+3))),
x = 1 .. (N-4)/8)}:
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MATHEMATICA
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M = 1000;
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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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