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%I #24 Apr 23 2024 11:21:19
%S 0,8,20,56,92,152,212,296,380,488,596,728,860,1016,1172,1352,1532,
%T 1736,1940,2168,2396,2648,2900,3176,3452,3752,4052,4376,4700,5048,
%U 5396,5768,6140,6536,6932,7352,7772,8216,8660,9128,9596,10088,10580,11096,11612,12152
%N a(n) is the number of permutable pieces in a standard n X n X n Rubik's cube.
%H <a href="/index/Ru#Rubik">Index entries for sequences related to Rubik cube</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = 8 + 12*(n-2) + 6*((n-2)^2 - (n mod 2)) for n > 1, a(1) = 0.
%F G.f.: 4*x^2*(x^3-4*x^2-x-2)/((x+1)*(x-1)^3).
%F a(n) = A005897(n-1) - A010677(n) for n>=2.
%F 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
%e 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.
%o (Python)
%o N = 20
%o seq = [0]
%o for n in range(2, N+1):
%o seq.append( 8 + 12*(n-2) + 6*((n-2)**2 - (n%2)) )
%Y Cf. A005897, A010677.
%K nonn,easy
%O 1,2
%A _William Riley Barker_, Jan 30 2023
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