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

%I A387286 #45 Sep 04 2025 00:34:10
%S A387286 0,16,132,504,1360,3000,5796,10192,16704,25920,38500,55176,76752,
%T A387286 104104,138180,180000,230656,291312,363204,447640,546000,659736,
%U A387286 790372,939504,1108800,1300000,1514916,1755432,2023504,2321160,2650500,3013696,3412992,3850704,4329220,4851000,5418576,6034552,6701604,7422480
%N A387286 Number of 2 X 2 square tiles in a discrete 4-dimensional hypercube of side length n.
%C A387286 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.
%C A387286 In 4D, the hypercube is interpreted inductively as n 3D cubes arranged in a row along the fourth axis ("moving-cube" model).
%C A387286 Equivalently, this sequence counts the 2X2 tiles in the 3D projection ("shadow") of the 4D hypercube.
%H A387286 Salvatore Ferraro, <a href="https://doi.org/10.5281/zenodo.16939782">Number of tiles in a discrete 4D hypercube</a>, Zenodo, 2025.
%H A387286 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A387286 a(n) = (n - 1)^2 * (3*n^2 + 2*n) = 3*n^4 - 4*n^3 - n^2 + 2*n.
%F A387286 G.f.: 4*x^3*(4 + 13*x + x^2)/(1 - x)^5. - _Stefano Spezia_, Aug 25 2025
%e A387286 a(2) = 16, a(3) = 132, a(4) = 504
%p A387286 a := n -> (n-1)^2*(3*n^2 + 2*n):
%p A387286 seq(a(n), n=1..40);
%t A387286 a[n_] := (n - 1)^2*(3 n^2 + 2 n); Table[a[n], {n, 1, 40}]
%o A387286 (Python)
%o A387286 def a(n):
%o A387286     return (n - 1)**2 * (3*n**2 + 2*n)
%o A387286 print([a(n) for n in range(1, 41)])
%Y A387286 Cf. A000290 (squares, 2D case), A270205 (3D case).
%K A387286 nonn,easy,new
%O A387286 1,2
%A A387286 _Salvatore Ferraro_, Aug 25 2025