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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, 1, 3, 12, 46, 163, 405, 770, 1252, 1921, 2816, 3977, 5464, 7313
OFFSET
1,4
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
Sequence in context: A026559 A188949 A184699 * A047013 A370478 A108368
KEYWORD
nonn
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