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

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

Offset: 1

Author

Nicolay Avilov, Jun 07 2023

Keywords

• nonn
• easy

Comments

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.

Examples

			a(3) = 6*8 - 12*1 - 2*8 = 20;
a(5) = 6*17 - 12*3 - 2*8 = 50.
		

Links

• Nicolay Avilov, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

Cf. A194274.

Programs

Mathematica

Join[{1},LinearRecurrence[{3,-4,4,-3,1},{8,20,32,50,80},51]] (* Stefano Spezia, Jun 08 2023 *)

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

Formula

a(n) = 6*A194274 - 12*n + 8, where n>1.

From Stefano Spezia, Jun 08 2023: (Start)

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)