A075886 Number of cubes at generation n when building fractal cube with edge ratio of 1/2.

1, 6, 30, 150, 726, 3510, 16854, 80886, 387606, 1857078, 8894550, 42598518, 204000918, 976929462, 4678286550, 22403128566, 107282481942, 513746046774, 2460185076054, 11781130000758, 56416485185430, 270162504104886

Offset: 1

Author

Gregory V. Richardson, Oct 17 2002

Keywords

• nonn
• easy

Comments

This sequence gives the number of cubes in each generation "#n" of the fractal cube with edge ratio of 1/2. To create the fractal cube, start with a cube of edge length 1 (the mother cube). Now attach 6 smaller cubes of edge length 1/2 to each free face of the mother cube. Keep edges parallel and place the smaller cubes centrally on each face of the mother cube. These 6 cubes are generation #2.

Next place 30 cubes of edge length 1/4 onto each free face of the 6 generation #2 cubes. Continue to place smaller cubes on each remaining free face. However, the generation #4 cubes touch the mother cube face-to-face such that some of the 5th generation cubes lie totally submerged within the mother cube. In later generations other cubes are submerged within cubes from their previous generations, too. Submerged cubes are not permitted, so these are not included in the totals.

The final displaced volume of the fractal cube (with R = 1/2) as n -> infinity = 2 22/23. If the intersections ARE allowed, the final total volume of all cubes is 3. In addition to the mother cube, there are 5 other types of cubes based on how many larger and smaller cubes are adjacent. The mother cube gives birth to 6 "A" cubes. An "A" cube begets another "A" and 4 "B" cubes. A "B" cube begets an "A", 3 "B" cubes and a "C". A "C" begets 3 "D" cubes and an "E". A "D" begets an "A", 2 "B" cubes and 2 "C" cubes. An "E" cube begets 2 "D" cubes and 2 "E" cubes.

Summing these up, the new number of "A" cubes = A + B + D. New "B" cubes = 4*A + 3*B + 2*D. New "C" cubes = B + 2*D. New "D" cubes = 3*C + 2*E. New "E" cubes = C + 2*E. In the final fractal cube, an "A" cube abuts one larger cube and its children abut no larger cubes. A "B" cube abuts one larger cube and its children abut one larger cube. A "C" cube abuts two larger cubes and its children abut one larger cube. A "D" cube abuts one larger cube and its children abut two larger cubes.

An "E" cube abuts two larger cubes and its children abut two larger cubes. The final fractal cube appears similar to a regular octahedron but apparently exhibits small indentations that differentiate it therefrom.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,4,-24).

Programs

Mathematica

Rest[CoefficientList[Series[x (-4x^2+x+1)/(24x^3-4x^2-5x+1),{x,0,30}],x]] (* or *) LinearRecurrence[{5,4,-24},{1,6,30},30] (* Harvey P. Dale, Mar 17 2012 *)

UBASIC

50 word *: point 80
100 cls: print: print "Fractal Cube Ratio = 1/2": print
110 locate 3: print "1";: locate 9: print "1"
120 locate 3: print "2";: locate 9: print "6"
130 A=6: B=0: C=0: D=0: E=0: G=2: Cc=2: V=1.75: Dv=1/8
140 G=G+1: Cc=Cc+1: Dv=Dv/8
150 Na=A+B+D: Nb=4*A+3*B+2*D
160 Nc=B+2*D: Nd=3*C+2*E: Ne=C+2*E
170 Ncubes = Na+Nb+Nc+Nd+Ne
180 A=Na: B=Nb: C=Nc: D=Nd: E=Ne: V=V+Ncubes*Dv
190 Locate 2: print G;: locate 8: print Ncubes
200 if Cc<40 then 140
210 print: locate 4: print "V = ";: print using(2,60),V: print
220 print: print " Push 'S' to stop or space bar to continue": print
230 A$=inkey: if A$="" then 230
240 if A$="s" or A$="S" then print: end
250 Cc=0: goto 140

Formula

G.f.: x(-4*x^2 + x + 1)/(24*x^3 - 4*x^2 - 5*x + 1). - Ralf Stephan, May 09 2004

a(n) = 5*a(n-1) + 4*a(n-2) - 24*a(n-3). - Harvey P. Dale, Mar 17 2012