A242526 Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at most 4.

1, 1, 1, 3, 12, 36, 90, 214, 521, 1335, 3473, 9016, 23220, 59428, 152052, 389636, 999776, 2566517, 6586825, 16899574, 43352560, 111213798, 285319258, 732016006, 1878072638, 4818362046, 12361809384, 31714901077, 81366445061, 208750870961

Offset: 1

Author

Stanislav Sykora, May 27 2014

Keywords

• nonn

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.

Examples

			The 3 cycles of length n=4 are: {1,2,3,4},{1,2,4,3},{1,3,2,4}.
The first and the last of the 1335 such cycles of length n=10 are:
C_1={1,2,3,4,6,7,8,10,9,5}, C_1335={1,4,8,10,9,7,6,3,2,5}.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100
• S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.

Crossrefs

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.

Programs

Mathematica

A242526[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
lpf[x_] := Length[Select[Abs[Differences[x]], # > 4 &]];
Join[{1, 1}, Table[A242526[n], {n, 3, 10}]]
(* OR, a less simple, but more efficient implementation. *)
A242526[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[Abs[First[perm] - Last[perm]] <= 4, 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] > 4, Continue[]];
      A242526[n, Join[perm, {new}],
       Complement[Range[2, n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Join[{1, 1}, Table[ct = 0; A242526[n, {1}, Range[2, n]]/2, {n, 3, 12}] ](* Robert Price, Oct 25 2018 *)

Formula

From Andrew Howroyd, Apr 08 2016: (Start)

Empirical: a(n) = 2*a(n-1) + a(n-2) - a(n-4) + 9*a(n-5) + 5*a(n-6) - a(n-7) - 7*a(n-8) - 10*a(n-9) + 2*a(n-10) + 2*a(n-11) + 2*a(n-12) + 4*a(n-13) - 2*a(n-17) - a(n-18) for n>20.

Empirical g.f.: x + (3 - 6*x - 2*x^2 - x^3 + 3*x^4 - 22*x^5 - 5*x^6 + x^7 + 8*x^8 + 14*x^9 - 6*x^10 + 2*x^11 - 6*x^12 - 6*x^13 - 3*x^15 + x^16 + 3*x^17) / (1 - 2*x - x^2 + x^4 - 9*x^5 - 5*x^6 + x^7 + 7*x^8 + 10*x^9 - 2*x^10 - 2*x^11 - 2*x^12 - 4*x^13 + 2*x^17 + x^18). (End)

Extensions

a(22)-a(30) from Andrew Howroyd, Apr 08 2016