A098928 Number of cubes that can be formed from the points of a cubical grid of n X n X n points.

0, 1, 9, 36, 100, 229, 473, 910, 1648, 2795, 4469, 6818, 10032, 14315, 19907, 27190, 36502, 48233, 62803, 80736, 102550, 128847, 160271, 197516, 241314, 292737, 352591, 421764, 501204, 592257, 696281, 814450, 948112, 1098607, 1267367

Offset: 1

Author

Ignacio Larrosa Cañestro, Oct 19 2004, Sep 29 2009

Keywords

• nonn

Comments

Skew cubes are allowed.

Examples

			For n = 3 there are 8 cubes of volume 1 and 1 cube of volume 8; thus a(3)=9. - _José María Grau Ribas_, Mar 15 2014
a(6)=229 because we can place 15^2 cubes in a 6 X 6 X 6 cubical grid with their edges parallel to the faces of the grid, plus 4 cubes of edge 3 with a vertex in each face of the lattice and the other two vertices on a diagonal.
		

Links

• Baitian Li, Table of n, a(n) for n = 1..10000 (terms 1..101 from E. J. Ionascu and R. A. Obando)
• E. J. Ionascu and R. A. Obando, Counting all cubes in {0,1,...,n}^3, arXiv:1003.4569 [math.NT], 2010.
• Eugen J. Ionascu and Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.
• Eugen J. Ionascu and R. A. Obando, Cubes in {0,1,...,N}^3, INTEGERS, 12A (2012), #A9. - From _N. J. A. Sloane_, Feb 05 2013
• I. Larrosa, SMSU Problem Corner.
• Baitian Li, C++ program for A098928

Crossrefs

Cf. A103158.

Cf. A000537 (without skew cubes), A002415 (number of squares with corners on an n X n grid), A108279, A102698.

Programs

Mathematica

Needs["Quaternions`"];
(* Initialize variables *)
R = 20;
NN = 1010;
(* Quaternion operations *)
test[q_Quaternion] :=
  Module[{unit, res, a, b, c, u, v, w, p},
   If[Round[Norm[q]] > R, Return[]];
   If[q == Quaternion[0, 0, 0, 0], Return[]];
   unit = Quaternion[0, 1, 0, 0];
   res = q ** unit ** Conjugate[q];
   a = Abs[res[[2]]] + Abs[res[[3]]] + Abs[res[[4]]];
   unit = Quaternion[0, 0, 1, 0];
   res = q ** unit ** Conjugate[q];
   b = Abs[res[[2]]] + Abs[res[[3]]] + Abs[res[[4]]];
   unit = Quaternion[0, 0, 0, 1];
   res = q ** unit ** Conjugate[q];
   c = Abs[res[[2]]] + Abs[res[[3]]] + Abs[res[[4]]];
   For[i = 1, i <= (R - 1)/Max[a, b, c], i++,
    If[SquareFreeQ[i], {u = a*i;
      v = b*i;
      w = c*i;
      p = Max[u, v, w] + 1;
      coe[[p + 1, 4]] += (1);
      coe[[p + 1, 3]] -= (u + v + w);
      coe[[p + 1, 2]] += (u*v + v*w + w*u);
      coe[[p + 1, 1]] -= (u*v*w)}]]];
(* Set up coefficient matrix *)
coe = ConstantArray[0, {NN, 4}];
(* Loop through quaternions *)
rt = Ceiling[Sqrt[R]] + 1;
For[s = -rt, s <= rt, s++,
  For[x = -rt, x <= rt, x++,
   For[y = -rt, y <= rt, y++,
    For[z = -rt, z <= rt, z++, test[Quaternion[s, x, y, z]];
     test[Quaternion[s + 0.5, x + 0.5, y + 0.5, z + 0.5]];]]]];
newCoe = coe;
newCoe[[2 ;; ;; 2]] = coe[[2 ;; ;; 2]]/2;
(* Calculate and output results *)
For[i = 2, i <= R + 1, i++, ans = 0;
  For[j = 4, j >= 1, j--, newCoe[[i, j]] += newCoe[[i - 1, j]];
   ans = ans*(i - 1) + newCoe[[i, j]];
   ];
  Print[i - 1, " ", ans/24];];
(* Haomin Yang, Aug 29 2023 *)

Extensions

Edited by Ray Chandler, Apr 05 2010

Further edited by N. J. A. Sloane, Mar 31 2016