A383585 Number of vertices of even degree in a cubic lattice n X n X n.

0, 0, 13, 32, 63, 112, 185, 288, 427, 608, 837, 1120, 1463, 1872, 2353, 2912, 3555, 4288, 5117, 6048, 7087, 8240, 9513, 10912, 12443, 14112, 15925, 17888, 20007, 22288, 24737, 27360, 30163, 33152, 36333, 39712, 43295, 47088, 51097, 55328, 59787, 64480, 69413, 74592, 80023, 85712, 91665, 97888

Offset: 1

Author

Nicolay Avilov, May 01 2025

Keywords

• nonn
• easy

Comments

An n X n X n cubic lattice is a graph with n^3 vertices. All interior vertices of the n X n X n cubic lattice are vertices of even degree, and all interior edge vertices of the cube are also vertices of even degree, so the number of vertices of even degree in an n X n X n lattice is (n - 2)^3 + 12*(n - 2).

Examples

			a(4) = 2^3 + 12*2 = 32;
a(5) = 3^3 + 12*3 = 63.
		

Links

• Nicolay Avilov, A technique for counting
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000578.

Programs

Mathematica

a[n_] := n^3 - 6*n^2 + 24*n - 32; a[1] = 0; Array[a, 48] (* Amiram Eldar, May 13 2025 *)

Formula

a(n) = n^3 - 6*n^2 + 24*n - 32 for n >= 2.

From Chai Wah Wu, May 19 2025: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5.

G.f.: x^3*(13*x^2 - 20*x + 13)/(x - 1)^4. (End)