A388140 Number of axis-aligned 2 X 2 squares in the 3D projection of a 3n-dimensional hypercube with side length 2.

6, 108, 1176, 10800, 92256, 762048, 6193536, 49939200, 401081856, 3214937088, 25744644096, 206057779200, 1648864813056, 13192528969728, 105546673913856, 844399160524800, 6755296362233856, 54042783212371968, 432343914961698816, 3458757916753920000, 27670089722291552256, 221360823331410935808

Offset

1,1

Comments

This is the case n = 2 of the parametric family A_{3k}(n) = 3*n^k*(n^k - 1)^2, which counts axis-aligned 2 X 2 squares in the 3D (orthogonal) projection of a d-dimensional hypercube with d = 3k. Thus a(n) = 3*2^n*(2^n - 1)^2 is the number of 2 X 2 tiles in the 3D projection of the 6, 9, 12, ...-dimensional hypercubes (side 2).

Links

Table of n, a(n) for n=1..22.

Salvatore Ferraro, Counting axis-aligned 2X2 tiles in the 3D projection of hypercubes (short note), Zenodo, 2025.

Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Formula

a(n) = 3*2^n*(2^n - 1)^2 = 3*2^(3*n) - 6*2^(2*n) + 3*2^n.

a(n) = A270205(2^n).

From Elmo R. Oliveira, Mar 20 2026: (Start)

G.f.: -6*x*(4*x + 1)/((2*x - 1)*(4*x - 1)*(8*x - 1)).

E.g.f.: 3*exp(2*x)*(1 - 2*exp(2*x) + exp(6*x)).

a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3) for n > 3. (End)

Maple

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

Mathematica

a[n_] := 3*2^n*(2^n - 1)^2;
Array[a, 40]

Prog

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

Crossrefs

Cf. A000079 (powers of 2), A000225 (Mersenne numbers), A000290 (squares).

Cf. A270205 (2 X 2 squares in an n X n X n cube).

Cf. A387286 (2 X 2 tiles in the 4D hypercube / 3D projection).

Sequence in context: A127213 A129003 A099138 * A230540 A295791 A361728

Adjacent sequences: A388137 A388138 A388139 * A388141 A388142 A388143

Keyword

nonn,easy

Author

Salvatore Ferraro, Sep 15 2025

Status

approved