cp's OEIS Frontend

A193416 Minimum surface area of polycubes with volume n.

%I A193416 #63 Feb 16 2025 08:33:15
%S A193416 6,10,14,16,20,22,24,24,28,30,32,32,36,38,40,40,42,42,46,48,50,50,52,
%T A193416 52,54,54,54,58,60,62,62,64,64,66,66,66,70,72,74,74,76,76,78,78,78,80,
%U A193416 80,80,84,86,88,88,90,90,92,92,92,94,94,94,96,96,96,96,100
%N A193416 Minimum surface area of polycubes with volume n.
%C A193416 First differences are 0, 2, or 4. - _Charles R Greathouse IV_, Aug 25 2011
%C A193416 From _Juan Barajas Martin_, Aug 27 2011 - Sep 03 2011, Sep 26 2001: (Start)
%C A193416 Initial cubic shape polycube edge = e; volume = e^3; minimum surface area = 6e^2.
%C A193416 Target cubic shape polycube edge = e+1; volume = (e+1)^3; minimum surface area = 6(e+1)^2.
%C A193416 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.
%C A193416 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.
%C A193416 Every layer is constructed as an Ulam spiral, starting at the center and until the completion of the layer.
%C A193416 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).
%C A193416 All other positions in the Ulam spiral will increase no area to the previous polycube.
%C A193416 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.
%C A193416 (End)
%H A193416 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polycube.html">Polycube</a>
%H A193416 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSpiral.html">Ulam Spiral</a>
%F A193416 a(n^3) = 6n^2, a(n) ~ 6n^(2/3). - _Charles R Greathouse IV_, Aug 25 2011
%F A193416 From _Juan Barajas Martin_, Aug 28 2011: (Start)
%F A193416 The following formula is derived from the Mathematica program below:
%F A193416 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]}
%F A193416 az[n_]:Floor[Power[n-1, (3)^-1]]
%F A193416 kz[n_]:=n-az(n)^3
%F A193416 c1-c3: number of unit cubes increasing the surface area by 2 in every layer (see Comments above). (End)
%F A193416 a(n) = 6*n - 2*A007818(n). - _Mohammed Yaseen_, Aug 08 2021
%e A193416 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
%t A193416 vals=100;
%t A193416 az[n_]:=Floor[(n-1)^(1/3)];
%t A193416 kz[n_]:=n-az[n]^3;
%t A193416 av[n_]:=6*az[n]^2;
%t A193416 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]]];
%t A193416 pz[n_]:=If[kz[n]<az[n]^2,kz[n],az[n]^2];
%t A193416 qz[n_]:=If[kz[n]<az[n]^2+(az[n]+1)*az[n],kz[n]-pz[n],az[n]*(az[n]+1)];
%t A193416 rz[n_]:=If[kz[n]<(az[n]+1)^2+az[n]*(az[n]+1)+az[n]^2,kz[n]-pz[n]-qz[n],(az[n]+1)^2];
%t A193416 f[j_]:=Ceiling[j/2]*Floor[j/2]+1;
%t A193416 Do[c1[g]=0;p=0;c1[1]=0;Do[If[f[j]<=pz[g],p++],{j,2,pz[g]}];c1[g]=p;,{g,2,vals}];
%t A193416 Do[c2[g]=0;p=0;c2[1]=0;Do[If[f[j]<=qz[g],p++],{j,2,qz[g]}];c2[g]=p;,{g,2,vals}];
%t A193416 Do[c3[g]=0;p=0;c3[1]=0;Do[If[f[j]<=rz[g],p++],{j,2,rz[g]}];c3[g]=p;,{g,2,vals}];
%t A193416 cv[n_]:=If[n>1,2*(c1[n]+c2[n]+c3[n]),2];
%t A193416 smin[n_]:=av[n]+bv[n]+cv[n];
%t A193416 Table[smin[n], {n,vals}] (* _Juan Barajas Martin_, Sep 01 2011 *)
%Y A193416 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), A007818.
%K A193416 nonn
%O A193416 1,1
%A A193416 _Juan Barajas Martin_, Jul 25 2011