A387286 Number of 2 X 2 square tiles in a discrete 4-dimensional hypercube of side length n.

0, 16, 132, 504, 1360, 3000, 5796, 10192, 16704, 25920, 38500, 55176, 76752, 104104, 138180, 180000, 230656, 291312, 363204, 447640, 546000, 659736, 790372, 939504, 1108800, 1300000, 1514916, 1755432, 2023504, 2321160, 2650500, 3013696, 3412992, 3850704, 4329220, 4851000, 5418576, 6034552, 6701604, 7422480

Offset: 1

Author

Salvatore Ferraro, Aug 25 2025

Keywords

• nonn
• easy
• new

Comments

This generalizes the 2D case, where an n X n grid has (n-1)^2 tiles, and the 3D case, where an n X n X n cube has 3n^3 - 6n^2 + 3n tiles.

In 4D, the hypercube is interpreted inductively as n 3D cubes arranged in a row along the fourth axis ("moving-cube" model).

Equivalently, this sequence counts the 2X2 tiles in the 3D projection ("shadow") of the 4D hypercube.

Examples

			a(2) = 16, a(3) = 132, a(4) = 504
		

Links

• Salvatore Ferraro, Number of tiles in a discrete 4D hypercube, Zenodo, 2025.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000290 (squares, 2D case), A270205 (3D case).

Programs

Maple

a := n -> (n-1)^2*(3*n^2 + 2*n):
seq(a(n), n=1..40);

Mathematica

a[n_] := (n - 1)^2*(3 n^2 + 2 n); Table[a[n], {n, 1, 40}]

Python

def a(n):
    return (n - 1)**2 * (3*n**2 + 2*n)
print([a(n) for n in range(1, 41)])

Formula

a(n) = (n - 1)^2 * (3*n^2 + 2*n) = 3*n^4 - 4*n^3 - n^2 + 2*n.

G.f.: 4*x^3*(4 + 13*x + x^2)/(1 - x)^5. - Stefano Spezia, Aug 25 2025