A389269 Number of axis-aligned 2 X 2 squares in the 3D projection of a (3n+1)-dimensional hypercube with side length 2.

0, 16, 240, 2464, 22080, 186496, 1532160, 12419584, 100008960, 802686976, 6431969280, 51497672704, 412149104640, 3297863827456, 26385594777600, 211095495245824, 1688806910853120, 13510627083943936, 108085703863173120, 864688379678162944, 6917518032528998400, 55340188240671932416

Offset

0,2

Comments

This is the case k = 2 of the parametric family A_{3n+1}(k) = (k^n-1)^2*k^(n+1)+2*k^n*(k^n - 1)*(k^(n+1)-1), which counts axis-aligned 2 X 2 squares in the 3D (orthogonal) projection of a d-dimensional hypercube with d = 3n+1. Thus a(n) = (2^n-1)^2*2^(n+1) +2*2^n*(2^n-1)*(2^(n+1)-1) is the number of 2 X 2 tiles in the 3D projection of the 4, 7, 10, ...-dimensional hypercubes (side 2).

Links

Table of n, a(n) for n=0..21.

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^(3*n+1) - 5*2^(2*n+1) + 2^(n+2).

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

G.f.: -16*x*(x + 1)/((8*x - 1)*(4*x - 1)*(2*x - 1)).

E.g.f.: 2*exp(2*x)*(2 - 5*exp(2*x) + 3*exp(6*x)).

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

Maple

a := n -> (2^n-1)^2*2^(n+1) +2*2^n*(2^n-1)*(2^(n+1)-1);
seq(a(n), n=0..39);

Mathematica

Table[(2^n - 1)^2*2^(n + 1) + 2*2^n*(2^n - 1)*(2^(n + 1) - 1), {n, 0, 39}]

Prog

(Python)
def a(n:int)->int:
    return (2**n-1)**2*2**(n+1) + 2*2**n*(2**n-1)*(2**(n+1) - 1)
print([a(n) for n in range(40)])

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: A283411 A161591 A376641 * A227440 A103975 A162791

Adjacent sequences: A389266 A389267 A389268 * A389270 A389271 A389272

Keyword

nonn,easy

Author

Salvatore Ferraro, Sep 27 2025

Status

approved