cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A284433 The number of positive integer sequences of length n with no duplicate substrings (forward or backward) of length greater than 1, every number different from its neighbors, and a minimal sum (= A282167(n)).

Original entry on oeis.org

1, 2, 6, 2, 18, 12, 24, 24, 1320, 240, 528, 720, 5664, 105888, 16992, 150048, 244800, 4266432, 4375296, 39598848, 50637312, 123310080, 202549248, 15925690368, 3765657600, 23454351360, 101739110400, 437749678080, 1572538613760, 11891771080704, 7862693068800, 185042766200832
Offset: 1

Views

Author

Peter Kagey, Mar 27 2017

Keywords

Examples

			For n = 4 the a(4) = 2 solutions are [1,3,2,1] and [1,2,3,1].

Crossrefs

Cf. A282167, A283558, A284431, A284432, A284436.

Extensions

Terms a(10) onward from Max Alekseyev, Feb 06 2025

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

Views

Author

Peter Kagey, Feb 07 2017

Keywords

Comments

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.

Examples

			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].

Crossrefs

Cf. A259280, A282167, A282168.
Cf. A282169 is the product analog.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

a(12)-a(21) from Lars Blomberg, Jun 10 2017
Terms a(22) onward from Max Alekseyev, Feb 04 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

Views

Author

Peter Kagey, Feb 07 2017

Keywords

Comments

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

Examples

			[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].

Crossrefs

Cf. A259280, A282166, A282167, A282193.

Extensions

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

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

Views

Author

Peter Kagey, Feb 07 2017

Keywords

Examples

			  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).

Crossrefs

Cf. A081456, A282164, A282169, A282170, A282193.
Cf. A282167 is the sum analog.

Extensions

Terms a(12) onward from Max Alekseyev, Feb 04 2025
Showing 1-4 of 4 results.

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