This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383585 #31 May 19 2025 11:42:05 %S A383585 0,0,13,32,63,112,185,288,427,608,837,1120,1463,1872,2353,2912,3555, %T A383585 4288,5117,6048,7087,8240,9513,10912,12443,14112,15925,17888,20007, %U A383585 22288,24737,27360,30163,33152,36333,39712,43295,47088,51097,55328,59787,64480,69413,74592,80023,85712,91665,97888 %N A383585 Number of vertices of even degree in a cubic lattice n X n X n. %C A383585 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). %H A383585 Nicolay Avilov, <a href="/A383585/a383585_4.jpg">A technique for counting</a> %H A383585 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1). %F A383585 a(n) = n^3 - 6*n^2 + 24*n - 32 for n >= 2. %F A383585 From _Chai Wah Wu_, May 19 2025: (Start) %F A383585 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5. %F A383585 G.f.: x^3*(13*x^2 - 20*x + 13)/(x - 1)^4. (End) %e A383585 a(4) = 2^3 + 12*2 = 32; %e A383585 a(5) = 3^3 + 12*3 = 63. %t A383585 a[n_] := n^3 - 6*n^2 + 24*n - 32; a[1] = 0; Array[a, 48] (* _Amiram Eldar_, May 13 2025 *) %Y A383585 Cf. A000578. %K A383585 nonn,easy %O A383585 1,3 %A A383585 _Nicolay Avilov_, May 01 2025