A259280 -id:A259280 - cp's OEIS frontend

A283558 The number of positive integer sequences of length n with no duplicate substrings of length greater than 1 and a minimal sum (= A259280(n)).

1, 1, 3, 2, 2, 6, 6, 48, 60, 168, 144, 288, 1872, 3744, 5760, 11520, 161280, 322560, 1866240, 2903040, 10782720, 8294400, 24883200, 282009600, 846028800, 3060633600, 9181900800, 10450944000, 31352832000, 668860416000, 1881169920000, 17850212352000, 41009504256000, 248816074752000, 381752082432000

Offset: 1

Peter Kagey, Mar 10 2017

nonn

			For n = 7 the a(7) = 6 sequences are
1,3,1,2,2,1,1;
1,2,2,1,3,1,1;
1,3,1,1,2,2,1;
1,1,3,1,2,2,1;
1,2,2,1,1,3,1; and
1,1,2,2,1,3,1.

A283557 is the product analog.

Cf. A259280, A284431, A284432, A284433.

s[1] = 1; s[n_] := Ceiling[(n+1+ Sum[Floor[Sqrt[2 k] + 1/2], {k, n-1}])/2]; subQ[w_] := Block[{n = Length@w}, Length@ Union@ Flatten[ Table[ Take[w, {i, j}], {j, 2, n}, {i, j - 1}], 1] == n (n-1)/2]; a[n_] := Sum[ Length@ Select[ Permutations@ e, subQ], {e, IntegerPartitions[ s[n], {n}]}]; Array[a, 10] (* Giovanni Resta, Mar 10 2017 *)

a(11)-a(13) from Giovanni Resta, Mar 10 2017

Terms a(14) onward from Max Alekseyev, Feb 06 2025

A274701 First differences of A259280.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8

Offset: 1

Braxton Carrigan, Peter Kagey, Joseph O'Brien, Jul 06 2016

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A259280, A185869, A002024.

If n is in A185869 then a(n) = ceiling(k) else a(n) = floor(k) where k is (A002024(n) + 1)/2.

A282164 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings of length greater than 1.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 12, 24, 48, 120, 240, 720, 1440, 4320, 8640, 25920, 60480, 181440, 483840, 1451520, 3870720, 11612160, 34836480, 104509440, 348364800, 1045094400, 3483648000, 10450944000, 38320128000, 114960384000, 459841536000, 1379524608000, 5518098432000, 16554295296000

Offset: 1

Peter Kagey, Braxton Carrigan, Feb 07 2017

nonn

a(n) = sqrt(A282165(n-1)) if A282165(n-1) is square.

			  a(1)  = 1   via [1];
  a(2)  = 1   via [1,1];
  a(3)  = 2   via [1,1,2];
  a(4)  = 2   via [1,1,2,1];
  a(5)  = 4   via [1,1,2,2,1];
  a(6)  = 6   via [1,1,2,1,3,1];
  a(7)  = 12  via [1,1,2,2,1,3,1];
  a(8)  = 24  via [1,1,2,1,3,1,4,1];
  a(9)  = 48  via [1,1,2,2,1,3,1,4,1];
  a(10) = 120 via [1,1,2,1,3,1,4,1,5,1].

Peter Kagey, Table of n, a(n) for n = 1..100

Product analog of A259280.

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

Peter Kagey, Feb 07 2017

nonn

For example, [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)  = 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].

Cf. A259280, A282167, A282168.

Cf. A282169 is the product analog.

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 *)

For n>=4, we seem to have a(n) = a(n-1) + a(n-2) - a(n-3) + d(n), where d(n) is either 0 or 1 (with a clear formula). This observation leads to the conjecture: for n>=4, a(n) = -3/2 + 2*n + n*m/2 - m*(2*m^2+15*m+46)/24 + (-1)^n*(m%2+2)/4 + (m%2)*3/8, where m is the largest integer such that (2*m^2 + 8*m + 1 + 3*(-1)^m)/4 <= n. - Max Alekseyev, May 28 2025

a(12)-a(21) from Lars Blomberg, Jun 10 2017

Terms a(22) onward from Max Alekseyev, Feb 04 2025

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

Peter Kagey, Feb 07 2017

nonn

For n>=7, we seem to have a(n) = a(n-1) + a(n-2) - a(n-3) + d(n), where d(n) is in {-2, 0, 2}. Compare to A282166. - Max Alekseyev, Jun 13 2025

			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].

Cf. A259280, A282166, A282168.

Terms a(11) onward from Max Alekseyev, Feb 05 2025

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

Peter Kagey, Feb 07 2017

nonn

This sequence shares first 12 terms with A025224, but then they diverge: a(13) = 33 > 32 = A025224(13).

We seem to have a(n) = a(n-1) + a(n-2) - a(n-3) + d(n), where d(n) is 0 or 1. Compare to A282166. - Max Alekseyev, Jun 13 2025

			[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].

Cf. A259280, A282166, A282167, A282193.

Edited and terms a(13) onward added by Max Alekseyev, Feb 05 2025

cp's OEIS Frontend

A283558 The number of positive integer sequences of length n with no duplicate substrings of length greater than 1 and a minimal sum (= A259280(n)).

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A274701 First differences of A259280.

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A282164 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings of length greater than 1.

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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.

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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.

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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.

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