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A259280 a(n) is the minimal sum of a positive integer sequence of length n with no duplicate substrings of length greater than 1. 7
1, 2, 4, 5, 7, 9, 11, 14, 16, 19, 21, 24, 27, 30, 33, 36, 40, 43, 47, 50, 54, 57, 61, 65, 69, 73, 77, 81, 85, 90, 94, 99, 103, 108, 112, 117, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 172, 177, 183, 188, 194, 199, 205, 210, 216, 221, 227, 233, 239, 245 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The lexicographically earliest positive integer sequence with no duplicate substrings is [1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, ...].

Note: Trivial substring of length 1 are allowed to recur, i.e., duplicate terms are permitted.

Non-examples of positive integer sequences with no duplicate substrings are

[1, 1, 1] (the substring [1, 1] occurs twice) and [1, 2, 3, 1, 2] (the substring [1, 2] occurs twice).

LINKS

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

FORMULA

a(1) = 1, a(n) = ceiling((n + 1 + A060432(n - 1))/2) for n > 1.

EXAMPLE

Lexicographically earliest examples:

a(1) = 1 via [1]

a(2) = 2 via [1, 1]

a(3) = 4 via [1, 1, 2]

a(4) = 5 via [1, 1, 2, 1]

a(5) = 7 via [1, 1, 2, 2, 1]

a(6) = 9 via [1, 1, 2, 1, 3, 1]

a(7) = 11 via [1, 1, 2, 2, 1, 3, 1]

a(8) = 14 via [1, 1, 2, 1, 3, 1, 4, 1]

a(9) = 16 via [1, 1, 2, 1, 3, 2, 2, 3, 1]

a(10) = 19 via [1, 1, 2, 1, 3, 2, 2, 3, 3, 1]

a(11) = 21 via [1, 1, 2, 1, 3, 2, 2, 3, 1, 4, 1]

a(12) = 24 via [1, 1, 2, 1, 3, 2, 2, 3, 3, 1, 4, 1]

a(13) = 27 via [1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 2, 4, 1]

PROG

(Ruby)

def a259280(n)

  lower_bound = 0.5 * (a060432(n - 1) + n + 1)

  lower_bound.ceil

end

CROSSREFS

Sequence in context: A258049 A057352 A156625 * A186388 A058577 A071917

Adjacent sequences:  A259277 A259278 A259279 * A259281 A259282 A259283

KEYWORD

nonn

AUTHOR

Peter Kagey, Nov 30 2015

STATUS

approved

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Last modified May 13 23:41 EDT 2021. Contains 343868 sequences. (Running on oeis4.)