A284435
The number of positive integer sequences of length n with no duplicate substrings (forward or backward) of length greater than 1 and a minimal product (= A282193(n)).
Original entry on oeis.org
1, 1, 2, 2, 4, 4, 4, 48, 144, 144, 144, 2304, 2160, 8640, 8640, 8640, 161280, 806400, 806400, 806400, 38534400, 39168000, 108864000, 108864000, 2794176000, 5370624000, 10741248000, 21482496000, 286355865600, 1002245529600, 2004491059200, 4008982118400, 61212572467200, 244850289868800, 489700579737600
Offset: 1
For n = 7 the a(7) = 4 solutions are:
[1,3,3,2,2,1,1],
[1,2,2,3,3,1,1],
[1,1,3,3,2,2,1], and
[1,1,2,2,3,3,1].
A282168
a(n) is the minimal sum 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, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 29, 33, 37, 41, 45, 49, 53, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 251, 258, 265, 273, 281, 289, 297, 305, 313, 321, 329, 337, 345, 353
Offset: 1
[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).
a(1) = 1 via [1];
a(2) = 2 via [1,1];
a(3) = 4 via [1,1,2];
a(4) = 6 via [1,1,2,2];
a(5) = 8 via [1,1,2,3,1];
a(6) = 10 via [1,1,2,2,3,1];
a(7) = 13 via [1,1,2,2,3,3,1];
a(8) = 16 via [1,1,2,2,3,1,4,2];
a(9) = 19 via [1,1,2,2,3,3,1,4,2];
a(10) = 22 via [1,1,2,2,3,1,4,2,5,1];
a(11) = 25 via [1,1,2,2,3,3,1,4,2,5,1];
a(12) = 29 via [1,1,2,2,3,3,1,4,4,2,5,1].
Edited and terms a(13) onward added by
Max Alekseyev, Feb 05 2025
A282169
a(n) is the minimal product 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, 2, 2, 6, 6, 24, 24, 120, 120, 576, 720, 2880, 4320, 17280, 30240, 120960, 241920, 967680, 1935360, 8709120, 17418240, 87091200, 174182400, 870912000, 1741824000, 9580032000, 19160064000, 104509440000, 229920768000, 1149603840000, 2759049216000, 13795246080000, 33108590592000, 165542952960000, 430411677696000
Offset: 1
[1,1] is not a valid sequence because 1 is self-adjacent.
[1,2,3,1,2] is not valid because the substring [1,2] appears twice.
a(1) = 1 via [1];
a(2) = 2 via [1,2];
a(3) = 2 via [1,2,1];
a(4) = 6 via [1,2,1,3];
a(5) = 6 via [1,2,1,3,1];
a(6) = 24 via [1,2,1,3,1,4];
a(7) = 24 via [1,2,1,3,1,4,1];
a(8) = 120 via [1,2,1,3,1,4,1,5];
a(9) = 120 via [1,2,1,3,1,4,1,5,1];
a(10) = 576 via [1,2,1,3,1,4,2,3,4,1];
a(11) = 720 via [1,2,1,3,1,4,1,5,1,6,1].
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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