A242525 - OEIS

%I #28 Jun 26 2022 17:31:19

%S 1,1,1,3,6,10,17,31,57,104,188,340,616,1117,2025,3670,6651,12054,

%T 21847,39596,71764,130065,235730,427238,774328,1403395,2543518,

%U 4609881,8354965,15142569,27444447,49740415,90149708,163387657,296124381,536696900

%N Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at most 3.

%C 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.

%H Andrew Howroyd, <a href="/A242525/b242525.txt">Table of n, a(n) for n = 1..100</a>

%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">On Neighbor-Property Cycles</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, 2014.

%F Empirical: a(n) = a(n-1)+a(n-2)+a(n-4)+a(n-5) for n>7. - _Andrew Howroyd_, Apr 08 2016

%F Empirical G.f.: x^2 + ((1-x)^2*(1+x)^2)/(1-x-x^2-x^4-x^5). - _Andrew Howroyd_, Apr 08 2016

%F Empirical first differences of A185265. - _Sean A. Irvine_, Jun 26 2022

%e For n=4, The three cycles are: C_1={1,2,3,4}, C_2={1,2,4,3}, C_3={1,3,2,4}.

%e The first and the last of the 104 such cycles of length n=10 are: C_1={1,2,3,5,6,8,9,10,7,4}, C_104={1,3,6,9,10,8,7,5,2,4}.

%t A242525[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;

%t j1f[x_] := Join[{1}, x, {1}];

%t lpf[x_] := Length[Select[Abs[Differences[x]], # > 3 &]];

%t Join[{1, 1}, Table[A242525[n], {n, 3, 10}]]

%t (* OR, a less simple, but more efficient implementation. *)

%t A242525[n_, perm_, remain_] := Module[{opt, lr, i, new},

%t    If[remain == {},

%t      If[Abs[First[perm] - Last[perm]] <= 3, ct++];

%t      Return[ct],

%t      opt = remain; lr = Length[remain];

%t      For[i = 1, i <= lr, i++,

%t       new = First[opt]; opt = Rest[opt];

%t       If[Abs[Last[perm] - new] > 3, Continue[]];

%t       A242525[n, Join[perm, {new}],

%t        Complement[Range[2, n], perm, {new}]];

%t       ];

%t      Return[ct];

%t      ];

%t    ];

%t Join[{1, 1},

%t Table[ct = 0; A242525[n, {1}, Range[2, n]]/2, {n, 3, 12}] ](* _Robert Price_, Oct 24 2018 *)

%o (C++) See the link.

%Y Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.

%K nonn

%O 1,4

%A _Stanislav Sykora_, May 27 2014

%E a(28)-a(35) from _Andrew Howroyd_, Apr 08 2016