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.
A007773(4)= 3: 0,2,1,3 -> 38; 0,1,2,3 -> 44; 0,1,3,2 ->46
A008782 := proc(n) local S,i,j,sumsq,npermut,p,per ; S := {} ; npermut := combinat[permute]([seq(i,i=0..n-1)]) ; for p from 1 to nops(npermut) do per := op(p,npermut) ; sumsq := 0 ; for i from 0 to n-1 do sumsq := sumsq + (add(op(1+(j mod n),per),j=i..i+2)) ^2 ; od ; S := S union {sumsq} ; od ; RETURN(nops(S)) ; end: for n from 1 to 20 do print(A008782(n)) ; od : # R. J. Mathar, Jun 18 2007
a[n_] := Module[{S, i, j, Sumsq, npermut, p, per}, S = {}; npermut = Permutations[Range[0, n-1]]; For[p = 1, p <= Length[npermut], p++, per = npermut[[p]]; Sumsq = 0; For[i = 0, i <= n-1, i++, Sumsq = Sumsq + Sum[per[[1+Mod[j, n] ]], {j, i, i+2}]^2]; S = S ~Union~ {Sumsq}]; Return[Length[S]]]; Table[Print[an = a[n]]; an, {n, 1, 10}] (* Jean-François Alcover, Jan 13 2014, translated from R. J. Mathar's Maple code *)
Consider n = 5: and the circular arrangements of {0,1,2,3,4}. Here are the values of [ A, B, C, D, E ] (A+B)^3 + (B+C)^3 +(C+D)^3 +(D+E)^3 +(E+A)^3:
[0,1,2,3,4], (0+1)^3 + (1+2)^3 +(2+3)^3 +(3+4)^3 +(4+0)^3 = 560;
[0,1,2,4,3], (0+1)^3 + (1+2)^3 +(2+4)^3 +(4+3)^3 +(3+0)^3 = 614;
[0,1,3,2,4], (0+1)^3 + (1+3)^3 +(3+2)^3 +(2+4)^3 +(4+0)^3 = 470;
[0,1,4,2,3], (0+1)^3 + (1+4)^3 +(4+2)^3 +(2+3)^3 +(3+0)^3 = 494;
[0,1,3,4,2], (0+1)^3 + (1+3)^3 +(3+4)^3 +(4+2)^3 +(2+0)^3 = 632;
[0,1,4,3,2], (0+1)^3 + (1+4)^3 +(4+3)^3 +(3+2)^3 +(2+0)^3 = 602;
[0,2,1,3,4], (0+2)^3 + (2+1)^3 +(1+3)^3 +(3+4)^3 +(4+0)^3 = 506;
[0,2,1,4,3], (0+2)^3 + (2+1)^3 +(1+4)^3 +(4+3)^3 +(3+0)^3 = 530;
[0,3,1,2,4], (0+3)^3 + (3+1)^3 +(1+2)^3 +(2+4)^3 +(4+0)^3 = 398;
[0,4,1,2,3], (0+4)^3 + (4+1)^3 +(1+2)^3 +(2+3)^3 +(3+0)^3 = 368;
[0,3,1,4,2], (0+3)^3 + (3+1)^3 +(1+4)^3 +(4+2)^3 +(2+0)^3 = 440;
[0,4,1,3,2], (0+4)^3 + (4+1)^3 +(1+3)^3 +(3+2)^3 +(2+0)^3 = 386;
There are 12 different values, so a(5) = 12.
A008781 := proc(n) local msu,p,c,i ; msu := {} ; for p in combinat[permute](n-1) do c := [0,op(p)] ; s := 0 ; for i from 0 to n-1 do s := s+(c[i+1]+c[1+modp(i+1,n)])^3 ; end do: msu := msu union {s} ; end do: nops(msu) ; end proc: # R. J. Mathar, Jul 18 2017
f[perm_] := Total[#]^3& /@ Partition[Join[{0}, perm, {0}], 2, 1] // Total;
a[n_] := a[n] = f /@ Permutations[Range[n - 1]] // Union // Length;
Reap[Do[Print[n, " ", a[n]]; Sow[a[n]], {n, 1, 12}]][[2, 1]] (* Jean-François Alcover, Feb 24 2020 *)