
EXAMPLE

a(3) = 4 since the permutation 1,2,3 does not meet the requirement (since 21 = 32) but the permutation 1,2,4 is okay as 21, 42, 41 are pairwise distinct.
a(4) = 6 since none of the permutations 1,2,4,5 and 1,4,2,5 meets the requirement (since 54 = 21 and 52 = 41), but the permutation 1,4,2,6 is okay as 41, 42, 62, 61 are pairwise distinct.
a(5) = 7 due to the permutation 1,6,2,4,7.
a(6) = 8 due to the permutation 1,4,6,2,7,8.
a(7) = 9 due to the permutation 1,4,8,6,7,2,9.
a(8) = 10 due to the permutation 1,7,4,9,8,6,2,10.
a(9) = 11 due to the permutation 1,6,7,9,2,10,4,8,11.
a(10) = 12 due to the permutation 1,6,8,9,2,11,7,10,4,12.
a(11) = 13 due to the permutation 1,12,2,11,4,10,6,9,7,8,13.
a(12) = 14 due to the permutation
1, 13, 2, 12, 4, 11, 6, 10, 7, 9, 8, 14.
a(13) = 15 due to the permutation
1, 11, 6, 8, 12, 9, 10, 2, 14, 7, 13, 4, 15.
a(14) = 16 due to the permutation
1, 12, 9, 8, 10, 15, 2, 11, 7, 13, 6, 14, 4, 16.
a(15) = 17 due to the permutation
1, 12, 9, 13, 4, 16, 6, 11, 10, 8, 14, 7, 15, 2, 17.
a(16) = 18 or 19 or 20 due to the permutation
1, 17, 2, 16, 4, 15, 6, 14, 7, 13, 8, 12, 9, 11, 10, 20.
Permutations for n = 13, 14, 15 were produced by QingHu Hou at Nankai Univ. on the author's request.
From Charlie Neder, Aug 23 2018: (Start)
a(16) = 18 due to the permutation
1, 11, 10, 12, 9, 13, 8, 14, 7, 15, 6, 17, 4, 16, 2, 18.
a(17) = 19 due to the permutation
1, 11, 12, 10, 13, 9, 14, 8, 15, 7, 16, 2, 18, 6, 17, 4, 19.
a(18) = 20 due to
1, 12, 11, 13, 10, 14, 9, 15, 8, 16, 7, 17, 2, 19, 6, 18, 4, 20. (End)
From Bert Dobbelaere, Sep 09 2019: (Start)
a(19) = 22 due to the permutation
1, 18, 2, 17, 8, 19, 7, 20, 6, 16, 9, 15, 10, 14, 11, 13, 12, 4, 22.
a(20) = 23 due to the permutation
1, 18, 7, 20, 6, 22, 4, 19, 9, 16, 8, 17, 11, 13, 12, 15, 10, 14, 2, 23.
a(21) = 24 due to the permutation
1, 19, 10, 20, 7, 18, 4, 16, 9, 17, 11, 15, 12, 14, 13, 8, 23, 6, 22, 2, 24.
a(22) = 25 due to the permutation
1, 22, 4, 23, 7, 24, 9, 18, 8, 20, 6, 19, 11, 15, 12, 17, 10, 16, 14, 13, 2, 25.
a(23) = 26 due to the permutation
1, 22, 7, 25, 6, 23, 10, 24, 4, 20, 8, 19, 11, 17, 13, 18, 9, 16, 14, 15, 12, 2, 26. (End)


MATHEMATICA

A program to find a(16) in terms of the values a(1), ..., a(15):
V[i_]:=V[i]=Part[Permutations[{2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}], i]
Do[Do[Do[If[Length[Union[{Abs[1Part[V[i], 1]]}, Table[Abs[Part[V[i], j]If[j<14, Part[V[i], j+1], n]], {j, 1, 14}]]]<15, Goto[aa]];
Print[n, " ", " ", V[i]]; Goto[bb]; Label[aa]; Continue, {i, 1, 14!}]; Continue, {n, 18, 20}]; Label[bb]; Break]
A228728[n_] := Module[{p, i, j, k, b, lim = 100},
If[n <= 2, A228728[n] = n,
j = A228728[n  1] + 1;
While[j < lim, A228728[n] = j;
p = Permutations[Table[A228728[k], {k, 2, n  1}]];
i = 1; While[i <= Length[p],
b = Join[{A228728[1]}, p[[i]], {A228728[n]}]; i++;
If[Length[Union[Join[Table[Abs[b[[k]]  b[[k + 1]]], {k, 1, n  1}], {Abs[b[[n]]  b[[1]]]}]]] == n, Return[j]]]; j++]]]
Table[A228728[n], {n, 1, 11}] (* Robert Price, Apr 04 2019 *)
