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A363518 Concentric square numbers on the faces of an n X n X n cube.

%I A363518 #15 Jul 15 2023 14:26:10
%S A363518 1,8,20,32,50,80,116,152,194,248,308,368,434,512,596,680,770,872,980,
%T A363518 1088,1202,1328,1460,1592,1730,1880,2036,2192,2354,2528,2708,2888,
%U A363518 3074,3272,3476,3680,3890,4112,4340,4568,4802,5048,5300,5552,5810,6080,6356,6632,6914,7208,7508,7808
%N A363518 Concentric square numbers on the faces of an n X n X n cube.
%C A363518 a(n) is the number of colored cubes in the outer layer of a cube made up of n^3 unit cubes. The cubes are painted in such a way that concentric square numbers are obtained on each face of the n X n X n cube.
%H A363518 Nicolay Avilov, <a href="/A363518/a363518.jpg">Illustration of initial terms</a>
%H A363518 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F A363518 a(n) = 6*A194274 - 12*n + 8, where n>1.
%F A363518 From _Stefano Spezia_, Jun 08 2023: (Start)
%F A363518 G.f.: (1 + 5*x + 5*x^4 + x^5)/((1 - x)^3*(1 + x^2)).
%F A363518 a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3)- 3*a(n-4) + a(n-5) for n > 6. (End)
%e A363518 a(3) = 6*8 - 12*1 - 2*8 = 20;
%e A363518 a(5) = 6*17 - 12*3 - 2*8 = 50.
%t A363518 Join[{1},LinearRecurrence[{3,-4,4,-3,1},{8,20,32,50,80},51]] (* _Stefano Spezia_, Jun 08 2023 *)
%o A363518 (Python)
%o A363518 def A363518(n): return 6*((3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2))-12*n+8 if n>1 else 1 # _Chai Wah Wu_, Jul 15 2023
%Y A363518 Cf. A194274.
%K A363518 nonn,easy
%O A363518 1,2
%A A363518 _Nicolay Avilov_, Jun 07 2023