The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”). Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A228772 Number of undirected circular permutations i_0,i_1,...,i_{n-1} of 0,1,...,n-1 such that i_0+i_1+i_2, i_1+i_2+i_3, ..., i_{n-3}+i_{n-2}+i_{n-1}, i_{n-2}+i_{n-1}+i_0, i_{n-1}+i_0+i_1 are pairwise distinct modulo n. 2
 0, 3, 2, 24, 24, 392, 513, 4080, 8090, 96816, 238296, 2023896, 7325520, 63277376, 277838352, 2185076682 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 COMMENTS Note that if n > 3 is not a multiple of 3 then a(n) > 0 since the natural circular permutation (0,1,2,...,n-1) meets the requirement. Conjecture: Let G be an additive abelian group. If G is cyclic or G contains no involution, then for any finite subset A of G with |A| = n > 3, there is a numbering a_1,...,a_n of the elements of A such that the n sums a_1+a+2+a_3, a_2+a_3+a_4, ..., a_{n-2}+a_{n-1}+a_n, a_{n-1}+a_n+a_1, a_n+a_1+a_2 are pairwise distinct. On Sep 13 2013, the author proved the conjecture for any torsion-free abelian group G. LINKS Zhi-Wei Sun, An additive theorem and restricted sumsets, Math. Res. Lett. 15(2008), 1263-1276. Zhi-Wei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014. EXAMPLE a(4) = 3 due to the circular permutations (0,1,2,3), (0,1,3,2) and (0,2,1,3). a(5) = 2 due to the circular permutations (0,1,2,3,4) and(0,2,4,1,3). a(6) > 0 due to the circular permutation (0,1,2,4,5,3). a(9) > 0 due to the circular permutation (0,1,2,3,8,5,6,7,4). MATHEMATICA (* A program to compute required circular permutations for n = 9. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0, 4, 7, 6, 5, 8, 3, 2, 1) is identical to (0, 1, 2, 3, 8, 5, 6, 7, 4) if we ignore direction.*) V[i_]:=Part[Permutations[{1, 2, 3, 4, 5, 6, 7, 8}], i] m=0 Do[If[Length[Union[Table[Mod[If[j==0, 0, Part[V[i], j]]+If[j<8, Part[V[i], j+1], 0]+If[j<7, Part[V[i], j+2], If[j==7, 0, Part[V[i], 1]]], 9], {j, 0, 8}]]]<9, Goto[aa]]; m=m+1; Print[m, ":", " ", 0, " ", Part[V[i], 1], " ", Part[V[i], 2], " ", Part[V[i], 3], " ", Part[V[i], 4], " ", Part[V[i], 5], " ", Part[V[i], 6], " ", Part[V[i], 7], " ", Part[V[i], 8]]; Label[aa]; Continue, {i, 1, 8!}] CROSSREFS Cf. A228626, A228766, A185645, A228728. Sequence in context: A151429 A151475 A105525 * A165714 A090883 A100645 Adjacent sequences:  A228769 A228770 A228771 * A228773 A228774 A228775 KEYWORD nonn,more,hard AUTHOR Zhi-Wei Sun, Sep 03 2013 EXTENSIONS a(10)-a(18) from Bert Dobbelaere, Sep 08 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 3 22:32 EST 2021. Contains 349468 sequences. (Running on oeis4.)