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

1,1

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A047845.

Sequence in context: A035511 A095035 A108027 * A117227 A360767 A366825

Adjacent sequences: A090764 A090765 A090766 * A090768 A090769 A090770

Keyword

nonn

Author

John H. Mason, Feb 02 2004

Extensions

More terms from Ray Chandler, Feb 04 2004

Status

approved