A284431
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 sum (= A282166(n)).
Original entry on oeis.org
1, 2, 1, 6, 2, 42, 12, 116, 18, 84, 168, 336, 384, 6864, 7872, 67392, 52800, 357120, 115200, 748800, 3093120, 11681280, 26853120, 43130880, 74649600, 2238382080, 3588157440, 51775856640, 63188398080, 653811056640, 480220876800, 4284050964480, 1316818944000, 11061279129600, 76921038028800
Offset: 1
For n = 5, the a(5) = 2 sequences are [1,3,1,2,1] and [1,2,1,3,1], each with a sum of A282166(5) = 8.
A282167
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, and no self-adjacent terms.
Original entry on oeis.org
1, 3, 6, 7, 11, 13, 17, 19, 25, 27, 31, 35, 39, 45, 47, 53, 57, 63, 67, 73, 77, 83, 87, 95, 99, 105, 111, 117, 123, 129, 135, 141, 149, 153, 161, 167, 175, 181, 189, 195, 203, 209, 217, 223, 231, 237, 247, 253, 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 351, 357, 367, 375, 385, 393, 403, 411, 421, 429, 439
Offset: 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).
a(1) = 1 via [1];
a(2) = 3 via [1,2];
a(3) = 6 via [1,2,3];
a(4) = 7 via [1,2,3,1];
a(5) = 11 via [1,2,3,1,4];
a(6) = 13 via [1,2,3,1,4,2];
a(7) = 17 via [1,2,3,1,4,5,1];
a(8) = 19 via [1,2,3,1,4,2,5,1];
a(9) = 25 via [1,2,3,1,4,2,5,1,6];
a(10) = 27 via [1,2,3,1,4,2,5,1,6,2].
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].
Showing 1-4 of 4 results.
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