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

%I A075886 #25 Jan 16 2024 17:33:21
%S A075886 1,6,30,150,726,3510,16854,80886,387606,1857078,8894550,42598518,
%T A075886 204000918,976929462,4678286550,22403128566,107282481942,513746046774,
%U A075886 2460185076054,11781130000758,56416485185430,270162504104886
%N A075886 Number of cubes at generation n when building fractal cube with edge ratio of 1/2.
%C A075886 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.
%C A075886 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.
%C A075886 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.
%C A075886 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.
%C A075886 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.
%H A075886 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,4,-24).
%F A075886 G.f.: x(-4*x^2 + x + 1)/(24*x^3 - 4*x^2 - 5*x + 1). - _Ralf Stephan_, May 09 2004
%F A075886 a(n) = 5*a(n-1) + 4*a(n-2) - 24*a(n-3). - _Harvey P. Dale_, Mar 17 2012
%t A075886 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 *)
%o A075886 (UBASIC)
%o A075886 50 word *: point 80
%o A075886 100 cls: print: print "Fractal Cube Ratio = 1/2": print
%o A075886 110 locate 3: print "1";: locate 9: print "1"
%o A075886 120 locate 3: print "2";: locate 9: print "6"
%o A075886 130 A=6: B=0: C=0: D=0: E=0: G=2: Cc=2: V=1.75: Dv=1/8
%o A075886 140 G=G+1: Cc=Cc+1: Dv=Dv/8
%o A075886 150 Na=A+B+D: Nb=4*A+3*B+2*D
%o A075886 160 Nc=B+2*D: Nd=3*C+2*E: Ne=C+2*E
%o A075886 170 Ncubes = Na+Nb+Nc+Nd+Ne
%o A075886 180 A=Na: B=Nb: C=Nc: D=Nd: E=Ne: V=V+Ncubes*Dv
%o A075886 190 Locate 2: print G;: locate 8: print Ncubes
%o A075886 200 if Cc<40 then 140
%o A075886 210 print: locate 4: print "V = ";: print using(2,60),V: print
%o A075886 220 print: print " Push 'S' to stop or space bar to continue": print
%o A075886 230 A$=inkey: if A$="" then 230
%o A075886 240 if A$="s" or A$="S" then print: end
%o A075886 250 Cc=0: goto 140
%K A075886 nonn,easy
%O A075886 1,2
%A A075886 _Gregory V. Richardson_, Oct 17 2002