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 A242532 Number of cyclic arrangements of S={2,3,...,n+1} such that the difference of any two neighbors is greater than 1, and a divisor of their sum. 16
 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 20, 39, 0, 0, 0, 0, 319, 967, 0, 0, 1464, 6114, 16856, 44370, 0, 0, 0, 0, 2032951, 8840796, 12791922, 101519154, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,14 COMMENTS a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S of n elements and a specific pair-property P. For more details, see the link and A242519. For this property P and sets {0,1,2,...,n-1} or {1,2,...,n} the problem does not appear to have any solution. a(40)=a(41)=a(42)=a(43)=a(46)=a(47)=0. - Fausto A. C. Cariboni, May 17 2017 LINKS Table of n, a(n) for n=1..37. S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014. EXAMPLE The shortest such cycle is of length n=9: {2,4,8,10,5,7,9,3,6}. The next a(n)>0 occurs for n=14 and has 20 solutions. The first and the last of these are: C_1={2,4,8,10,5,7,14,12,15,13,11,9,3,6}, C_2={2,4,12,15,13,11,9,3,5,7,14,10,8,6}. MATHEMATICA A242532[n_] := Count[Map[lpf, Map[j2f, Permutations[Range[3, n + 1]]]], 0]/2; j2f[x_] := Join[{2}, x, {2}]; dvf[x_] := Module[{i}, Table[Abs[x[[i]] - x[[i + 1]]] > 1 && Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i, Length[x] - 1}]]; lpf[x_] := Length[Select[dvf[x], ! # &]]; Table[A242532[n], {n, 1, 10}] (* OR, a less simple, but more efficient implementation. *) A242532[n_, perm_, remain_] := Module[{opt, lr, i, new}, If[remain == {}, If[Abs[First[perm] - Last[perm]] > 1 && Divisible[First[perm] + Last[perm], First[perm] - Last[perm]], ct++]; Return[ct], opt = remain; lr = Length[remain]; For[i = 1, i <= lr, i++, new = First[opt]; opt = Rest[opt]; If[Abs[Last[perm] - new] <= 1 || ! Divisible[Last[perm] + new, Last[perm] - new], Continue[]]; A242532[n, Join[perm, {new}], Complement[Range[3, n + 1], perm, {new}]]; ]; Return[ct]; ]; ]; Table[ct = 0; A242532[n, {2}, Range[3, n + 1]]/2, {n, 1, 15}] (* Robert Price, Oct 25 2018 *) PROG (C++) See the link. CROSSREFS Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242533, A242534. Sequence in context: A217427 A345339 A165442 * A132606 A054288 A238245 Adjacent sequences: A242529 A242530 A242531 * A242533 A242534 A242535 KEYWORD nonn,hard,more AUTHOR Stanislav Sykora, May 30 2014 EXTENSIONS a(29)-a(37) from Fausto A. C. Cariboni, May 17 2017 STATUS approved

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Last modified December 6 08:34 EST 2023. Contains 367600 sequences. (Running on oeis4.)