

A340458


Minimum length of the string over the alphabet of 3 or more symbols that has exactly n substring palindromes. Substrings are counted as distinct if they start at different offsets.


1



1, 2, 2, 3, 4, 3, 4, 5, 5, 4, 5, 6, 6, 7, 5, 6, 7, 7, 8, 8, 6, 7, 8, 8, 9, 9, 9, 7, 8, 9, 9, 10, 10, 10, 11, 8, 9, 10, 10, 11, 11, 11, 12, 12, 9, 10, 11, 11, 12, 12, 12, 13, 13, 14, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 14, 11, 12, 13, 13, 14, 14, 14, 15, 15, 16, 15, 16, 12, 13, 14, 14, 15, 15, 15, 16, 16, 17, 16, 17, 17
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OFFSET

1,2


COMMENTS

The uploaded Python script uses G. Manacher's algorithm to efficiently calculate the number of palindromes.


LINKS

Serguei Zolotov, Table of n, a(n) for n = 1..5000
Glenn Manacher, A new lineartime "online" algorithm for finding the smallest initial palindrome of a string, Journal of the ACM, (1975) 22 (3): 346351.
Serguei Zolotov, Table of n, a(n), sample string for n = 1..5000
Serguei Zolotov, Python script to generate bfile and afile


FORMULA

a(k*(k+1)/2) = k, from a string of k identical symbols.


EXAMPLE

The string AAA with length 3 has 6 palindromic substrings:
A starting at offset 1,
A starting at offset 2,
A starting at offset 3,
AA starting at offset 1,
AA starting at offset 2,
AAA starting at offset 1.
There is no shorter string with exactly 6 substring palindromes. So a(6) = 3.


CROSSREFS

Sequence in context: A112342 A256094 A063712 * A185977 A204006 A106251
Adjacent sequences: A340455 A340456 A340457 * A340459 A340460 A340461


KEYWORD

nonn


AUTHOR

Serguei Zolotov, Feb 13 2021


STATUS

approved



