A008781 For any circular arrangement of 0..n-1, let S be the sum of cubes of every sum of two contiguous numbers; then a(n) is the number of distinct values of S.

1, 1, 1, 3, 12, 46, 163, 405, 770, 1252, 1921, 2816, 3977, 5464, 7313

Offset

1,4

Links

Table of n, a(n) for n=1..15.

Example

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.

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A007773, A008782.

Sequence in context: A026559 A188949 A184699 * A047013 A370478 A108368

Adjacent sequences: A008778 A008779 A008780 * A008782 A008783 A008784

Keyword

nonn

Author

Olivier Gérard

Extensions

Corrected by Naohiro Nomoto, Sep 10 2001

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 29 2002

a(12) from Alois P. Heinz, May 26 2013

a(13)-a(15) from Sean A. Irvine, Apr 04 2018

Status

approved