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 A242526 Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at most 4. 16
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..100 S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014. 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) EXAMPLE 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}. 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 *) PROG (C++) See the link. CROSSREFS Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534. Sequence in context: A135190 A101069 A225259 * A167667 A292291 A215919 Adjacent sequences:  A242523 A242524 A242525 * A242527 A242528 A242529 KEYWORD nonn AUTHOR Stanislav Sykora, May 27 2014 EXTENSIONS a(22)-a(30) from Andrew Howroyd, Apr 08 2016 STATUS approved

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Last modified October 21 14:18 EDT 2019. Contains 328301 sequences. (Running on oeis4.)