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

1,1

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

Links

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

Wikipedia, Symmetric polynomial

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.

Example

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

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.

Prog

(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

Crossrefs

Cf. A047845, A090767.

Sequence in context: A346917 A343154 A048734 * A077534 A164998 A165000

Adjacent sequences: A277867 A277868 A277869 * A277871 A277872 A277873

Keyword

nonn

Author

Eric R. Carter, Nov 02 2016

Status

approved