A360222 a(n) is the number of permutable pieces in a standard n X n X n Rubik's cube.

0, 8, 20, 56, 92, 152, 212, 296, 380, 488, 596, 728, 860, 1016, 1172, 1352, 1532, 1736, 1940, 2168, 2396, 2648, 2900, 3176, 3452, 3752, 4052, 4376, 4700, 5048, 5396, 5768, 6140, 6536, 6932, 7352, 7772, 8216, 8660, 9128, 9596, 10088, 10580, 11096, 11612, 12152

Offset: 1

Author

William Riley Barker, Jan 30 2023

Keywords

• nonn
• easy

Examples

			The 2 X 2 X 2 Rubik's cube consists of 8 corner pieces, so a(2) = 8; the 3 X 3 X 3 cube has 8 corner pieces, 12 edge pieces, and 6 non-permutable center pieces, so a(3) = 8 + 12 = 20.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to Rubik cube
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A005897, A010677.

Programs

Mathematica

A360222[n_] := If[n == 1, 0, 6*((n-2)*n - Mod[n, 2]) + 8]; Array[A360222, 50] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {0, 8, 20, 56, 92}, 50] (* Paolo Xausa, Oct 04 2024 *)

Python

N = 20
seq = [0]
for n in range(2, N+1):
   seq.append( 8 + 12*(n-2) + 6*((n-2)**2 - (n%2)) )

Formula

a(n) = 8 + 12*(n-2) + 6*((n-2)^2 - (n mod 2)) for n > 1, a(1) = 0.

G.f.: 4*x^2*(x^3-4*x^2-x-2)/((x+1)*(x-1)^3).

a(n) = A005897(n-1) - A010677(n) for n>=2.

E.g.f.: 2*(2*(x - 2) + (3*x^2 - 3*x + 4)*cosh(x) + (3*x^2 - 3*x + 1)*sinh(x)). - Stefano Spezia, Feb 02 2023