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.

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

1,2

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

Links

Table of n, a(n) for n=1..64.

Example

[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.

Sequence in context: A130798 A165453 A025224 * A294023 A096182 A186347

Adjacent sequences: A282165 A282166 A282167 * A282169 A282170 A282171

Keyword

nonn

Author

Peter Kagey, Feb 07 2017

Extensions

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

Status

approved