The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A193416 Minimum surface area of polycubes with volume n. 0
 6, 10, 14, 16, 20, 22, 24, 24, 28, 30, 32, 32, 36, 38, 40, 40, 42, 42, 46, 48, 50, 50, 52, 52, 54, 54, 54, 58, 60, 62, 62, 64, 64, 66, 66, 66, 70, 72, 74, 74, 76, 76, 78, 78, 78, 80, 80, 80, 84, 86, 88, 88, 90, 90, 92, 92, 92, 94, 94, 94, 96, 96, 96, 96, 100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS First differences are 0, 2, or 4. - Charles R Greathouse IV, Aug 25 2011 From Juan Barajas Martin, Aug 27 2011 - Sep 03 2011, Sep 26 2001: (Start) Initial cubic shape polycube edge = e; volume = e^3; minimum surface area = 6e^2. Target cubic shape polycube edge = e+1; volume = (e+1)^3; minimum surface area = 6(e+1)^2. The target polycube is achieved by the successive addition of 3 orthogonal layers of unit cubes to the faces of the initial polycube. The number of unit cubes added in each successive layer take values e^2, e*(e+1) and (e+1)^2, with every layer fully completed before starting the following layer. This is so because the first unit cube added on every layer will add a surface area of 4 to the previous polycube, which is greater than keeping the new unit cube in the current layer, where we ensure that area will increase by 2 at maximum. Every layer is constructed as an Ulam spiral, starting at the center and until the completion of the layer. Except for the first cube placed on every layer which will increase the surface area of the previous polycube by 4, we can count 2 surface area increments for the unit cubes placed at the positions given by the function of the Quarter-squares + 1 sequence (i.e., A033638 = A002620 + 1, which set the locations of right angle turns in Ulam square spiral). All other positions in the Ulam spiral will increase no area to the previous polycube. An infinite sequence of polycubes with minimal surface area is approached between pairs of successive polycubes with cubic shape and edge values (e) and (e+1), having respectively polycube volumes e^3 and (e+1)^3 and minimum surface areas 6e^2 and 6(e+1)^2. (End) LINKS Eric Weisstein's World of Mathematics, Polycube Eric Weisstein's World of Mathematics, Ulam Spiral FORMULA a(n^3) = 6n^2, a(n) ~ 6n^(2/3). - Charles R Greathouse IV, Aug 25 2011 From Juan Barajas Martin, Aug 28 2011: (Start) The following formula is derived from the Mathematica program below: smin[n]={6 Floor[(-1+n)^(1/3)]^2+If[n==1,4,If[kz[n]>az[n]^2+az[n] (az[n]+1),12,If[kz[n]>az[n]^2,8,4]]]+If[n>1,2 (c1[n]+c2[n]+c3[n]),2]} az[n_]:Floor[Power[n-1, (3)^-1]] kz[n_]:=n-az(n)^3 c1-c3: number of unit cubes increasing the surface area by 2 in every layer (see Comments above). (End) EXAMPLE The unique polycube of volume 1 is a cube with surface area 6, so a(1) = 6. There are eight polycubes of volume 4, of which seven have surface area 18 and one has surface area 16, so a(4) = 16. - Charles R Greathouse IV, Aug 25 2011 MATHEMATICA vals=100; az[n_]:=Floor[(n-1)^(1/3)]; kz[n_]:=n-az[n]^3; av[n_]:=6*az[n]^2; bv[n_]:=If[n==1, 4, If[kz[n]>az[n]^2+(az[n]+1)*az[n], 12, If[kz[n]>az[n]^2, 8, 4]]]; pz[n_]:=If[kz[n]1, 2*(c1[n]+c2[n]+c3[n]), 2]; smin[n_]:=av[n]+bv[n]+cv[n]; Table[smin[n], {n, vals}] (* Juan Barajas Martin, Sep 01 2011 *) CROSSREFS Cf. A033638 (A002620 + 1) identify the unit cubes which will increase the minimum area by 2 (locations of right angle turns in Ulam square spiral). Sequence in context: A183072 A225704 A228301 * A315160 A075777 A315161 Adjacent sequences:  A193413 A193414 A193415 * A193417 A193418 A193419 KEYWORD nonn AUTHOR Juan Barajas Martin, Jul 25 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 17 19:51 EDT 2021. Contains 343070 sequences. (Running on oeis4.)