A284434
The number of positive integer sequences of length n with no duplicate substrings of length greater than 1, every number different from its neighbors, and a minimal product (= A282169(n)).
Original entry on oeis.org
1, 2, 1, 6, 2, 24, 6, 120, 24, 84, 192, 336, 360, 1680, 2160, 10080, 26640, 70560, 80640, 564480, 645120, 13789440, 10644480, 78382080, 43545600, 783820800, 435456000, 2264371200, 12454041600, 22643712000, 117747302400, 466460467200, 367873228800, 2391175987200, 4414478745600
Offset: 1
For n = 7 the a(7) = 6 solutions are:
[1,4,1,3,1,2,1],
[1,3,1,4,1,2,1],
[1,4,1,2,1,3,1],
[1,2,1,4,1,3,1],
[1,3,1,2,1,4,1], and
[1,2,1,3,1,4,1].
A282193
a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1.
Original entry on oeis.org
1, 1, 2, 4, 6, 12, 36, 96, 240, 480, 1440, 5760, 17280, 40320, 120960, 483840, 1935360, 5806080, 17418240, 69672960, 348364800, 1045094400, 3832012800, 15328051200, 76640256000, 229920768000, 919683072000, 4598415360000, 22072393728000, 71735279616000, 286941118464000, 1434705592320000
Offset: 1
a(1) = 1 via [1];
a(2) = 1 via [1,1];
a(3) = 2 via [1,1,2];
a(4) = 4 via [1,1,2,2];
a(5) = 6 via [1,1,2,3,1];
a(6) = 12 via [1,1,2,2,3,1];
a(7) = 36 via [1,1,2,2,3,3,1];
a(8) = 96 via [1,1,2,2,3,1,4,2];
a(9) = 240 via [1,1,2,2,3,1,4,5,1];
a(10) = 480 via [1,1,2,2,3,1,4,2,5,1];
a(11) = 1440 via [1,1,2,2,3,3,1,4,2,5,1];
a(12) = 5760 via [1,1,2,2,3,1,4,2,5,1,6,2].
...
[1,2,3,1,2] is invalid because the substring [1,2] appears twice.
[1,2,1] is invalid because the substring [1,2] appears twice (once forward and once backward).
A282166
a(n) is the minimal sum of a positive integer sequence of length n with no duplicate substrings of length greater than 1, and every number different from its neighbors.
Original entry on oeis.org
1, 3, 4, 7, 8, 12, 13, 17, 18, 22, 24, 28, 30, 35, 37, 42, 44, 49, 51, 56, 59, 64, 67, 72, 75, 81, 84, 90, 93, 99, 102, 108, 111, 117, 121, 127, 131, 137, 141, 147, 151, 158, 162, 169, 173, 180, 184, 191, 195, 202, 206, 213, 218, 225, 230, 237, 242, 249, 254, 261, 266, 274, 279, 287, 292, 300, 305, 313, 318, 326, 331, 339, 344, 352, 358, 366, 372, 380, 386, 394
Offset: 1
a(1) = 1 via [1];
a(2) = 3 via [1,2];
a(3) = 4 via [1,2,1];
a(4) = 7 via [1,2,1,3];
a(5) = 8 via [1,2,1,3,1];
a(6) = 12 via [1,2,1,3,1,4];
a(7) = 13 via [1,2,1,3,1,4,1];
a(8) = 17 via [1,2,1,3,1,4,2,3];
a(9) = 18 via [1,2,1,3,2,3,1,4,1];
a(10) = 22 via [1,2,1,3,1,4,2,3,4,1];
a(11) = 24 via [1,2,1,3,2,3,1,4,1,5,1].
-
Table[Module[{s = Select[Permutations[Range@ n - 1, n], Length@ # > 1 &]}, Total@ First@ MinimalBy[#, Total] &@ DeleteCases[#, w_ /; Apply[Times, If[Length@ # > 0, Rest@ #, #] &@ Union@ Map[SequenceCount[w, #] &, s]] > 1] &@ Apply[Join, Map[MinimalBy[#, Total] &, Table[Select[Tuples[Range@ k, n], Function[w, Times @@ Boole@ {Length@ Union@ w == k, First@ #, If[n > 2, Xor @@ Rest@ #, True]} == 1 &@ Map[Length@ Split@ # == Length@ # &, {w, w[[1 ;; -1 ;; 2]], Rest[w][[1 ;; -1 ;; 2]]}]]], {k, n}]]]], {n, 7}] (* Michael De Vlieger, Mar 27 2017, Version 10 *)
A282170
a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1, and no self-adjacent terms.
Original entry on oeis.org
1, 2, 6, 6, 24, 48, 120, 240, 1440, 2880, 10080, 20160, 120960, 322560, 1209600, 2903040, 17418240, 58060800, 174182400, 638668800, 3483648000, 15328051200, 38320128000, 199264665600, 919683072000, 4828336128000, 11955879936000, 71735279616000, 334764638208000, 1506440871936000, 5021469573120000, 30128817438720000
Offset: 1
a(1) = 1 via [1];
a(2) = 2 via [1,2];
a(3) = 6 via [1,2,3];
a(4) = 6 via [1,2,3,1];
a(5) = 24 via [1,2,3,1,4];
a(6) = 48 via [1,2,3,1,4,2];
a(7) = 120 via [1,2,3,1,4,5,1];
a(8) = 240 via [1,2,3,1,4,2,5,1];
a(9) = 1440 via [1,2,3,1,4,2,5,1,6];
a(10) = 2880 via [1,2,3,1,4,2,5,1,6,2];
a(11) = 10080 via [1,2,3,1,4,2,5,1,6,7,1].
Examples:
[1,1] is invalid because 1 is self-adjacent.
[1,2,3,1,2] is invalid because the substring [1,2] appears twice.
[1,2,1] is invalid because the substring [1,2] appears twice (once forward and once backward).
Showing 1-4 of 4 results.
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