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.

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

Offset: 1

Author

John H. Mason, Feb 02 2004

Keywords

• nonn

Comments

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.

Examples

			a(1) = 12 because there are 12 edges to a cube.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A047845.

Programs

Maple

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)}:
sort(convert(S,list)); # Robert Israel, Feb 18 2016

Mathematica

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 *)

Extensions

More terms from Ray Chandler, Feb 04 2004