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A363518
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Concentric square numbers on the faces of an n X n X n cube.
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1
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1, 8, 20, 32, 50, 80, 116, 152, 194, 248, 308, 368, 434, 512, 596, 680, 770, 872, 980, 1088, 1202, 1328, 1460, 1592, 1730, 1880, 2036, 2192, 2354, 2528, 2708, 2888, 3074, 3272, 3476, 3680, 3890, 4112, 4340, 4568, 4802, 5048, 5300, 5552, 5810, 6080, 6356, 6632, 6914, 7208, 7508, 7808
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = 6*A194274 - 12*n + 8, where n>1.
G.f.: (1 + 5*x + 5*x^4 + x^5)/((1 - x)^3*(1 + x^2)).
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)
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EXAMPLE
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a(3) = 6*8 - 12*1 - 2*8 = 20;
a(5) = 6*17 - 12*3 - 2*8 = 50.
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MATHEMATICA
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Join[{1}, LinearRecurrence[{3, -4, 4, -3, 1}, {8, 20, 32, 50, 80}, 51]] (* Stefano Spezia, Jun 08 2023 *)
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PROG
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(Python)
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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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