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 A228728 a(1)=1, a(2)=2 and for n > 2, a(n) is the least integer > a(n-1) such that there is a permutation b(1), ..., b(n) of a(1), ..., a(n) with b(1) = a(1) and b(n) = a(n), and with the n numbers |b(1)-b(2)|, |b(2)-b(3)|, ..., |b(n-1)-b(n)|, |b(n)-b(1)| pairwise distinct. 8
 1, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: For any n distinct real numbers a_1 < a_2 < ... < a_n, if there is a permutation b_1,b_2,...,b_n of a_1,...,a_n with |b_1-b_2|, |b_2-b_3|, ..., |b_{n-1}-b_n|, |b_n-b_1| pairwise distinct, then there exists a permutation c_1,c_2,...,c_n of a_1,...,a_n with c_1 = a_1 and c_n = a_n such that the n numbers |c_1-c_2|, |c_2-c_3|, ..., |c_{n-1}-c_n|, |c_n-c_1| are pairwise distinct. This conjecture is somewhat curious but we are unable to find a counterexample. LINKS Table of n, a(n) for n=1..23. Zhi-Wei Sun, A problem on circular permutations, a message to Number Theory List, Sep 01 2013. Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014. EXAMPLE a(3) = 4 since the permutation 1,2,3 does not meet the requirement (since 2-1 = 3-2) but the permutation 1,2,4 is okay as 2-1, 4-2, 4-1 are pairwise distinct. a(4) = 6 since none of the permutations 1,2,4,5 and 1,4,2,5 meets the requirement (since 5-4 = 2-1 and 5-2 = 4-1), but the permutation 1,4,2,6 is okay as 4-1, 4-2, 6-2, 6-1 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 Qing-Hu 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[1-Part[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 *) CROSSREFS Cf. A185645, A187815. Sequence in context: A272633 A240817 A174416 * A138888 A039240 A039183 Adjacent sequences: A228725 A228726 A228727 * A228729 A228730 A228731 KEYWORD nonn,more AUTHOR Zhi-Wei Sun, Aug 31 2013 EXTENSIONS a(16)-a(18) from Charlie Neder, Aug 23 2018 Name clarified by Robert Price, Apr 04 2019 a(19)-a(23) from Bert Dobbelaere, Sep 09 2019 STATUS approved

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