A277870 Ordered number of unit edges needed to build every 4-orthotope from hypercubes.

32, 52, 72, 84, 92, 112, 116, 132, 135, 148, 152, 160, 172, 180, 186, 192, 204, 212, 216, 232, 237, 244, 248, 252, 256, 260, 272, 276, 288, 292, 297, 308, 312, 316, 326, 332, 336, 339, 340, 352, 372, 378, 380, 384, 390, 392, 396, 404, 408, 412, 415, 424, 428

Offset: 1

Author

Eric R. Carter, Nov 02 2016

Keywords

• nonn

Comments

Ordered number of edges required to construct every hyperrectangle as a union of unit hypercubes. The sequence gives the n-th smallest such number, and generalizes the two-dimensional A047845 and the three-dimensional A090767 to four dimensions.

Does a(n) ~ n? - Charles R Greathouse IV, Nov 06 2016

Examples

			a(1)=32 as this is the number of edges in the unit hypercube.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Wikipedia, Symmetric polynomial

Crossrefs

Cf. A047845, A090767.

Programs

Mathematica

Edges[x_,y_,z_,w_]:=(4*x*y*z*w)+3*((w*x*z)+(w*y*z)+(w*x*y)+(x*y*z))+2*((w*x)+(w*y)+(w*z)+(x*y)+(x*z)+(y*z))+x+y+z+w;inputs=Tuples[Range[s],4];Union[Table[Edges[inputs[[k]][[1]],inputs[[k]][[2]],inputs[[k]][[3]],inputs[[k]][[4]]],{k,1,Length[inputs]}]]
Accuracy to 170 terms is achieved for s>=5764801, and for the entire list in the limit as s approaches infinity.

PARI

list(lim)=my(v=List()); for(w=1,(lim-12)\20, for(x=1, min((lim-8*w-4)\(12*w+8),w), for(y=1,min((lim-5*w*x-3*x-3*w-1)\(7*w*x+5*x+5*w+3),x), forstep(n=((7*w+5)*y+(5*w+3))*x+(5*w + 3)*y+3*w+1, lim, ((4*w+3)*y+3*w+2)*x+(3*w+2)*y+2*w+1, listput(v,n))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2016

Formula

These numbers are of the form: 4wxyz + 3(wxz+wyz+wxy+xyz) + 2(wx+wy+wz+xy+xz+yz) + w+x+y+z for any positive integers w, x, y, z.