A351490 Irregular triangle read by rows: T(n,k) is the minimum number of alphabetic symbols in a regular expression for the k lexicographically first palindromes of odd length 2*n-1 over a binary alphabet, n >= 1, 1 <= k <= 2^n.

1, 2, 3, 4, 7, 8, 5, 6, 9, 10, 15, 16, 19, 20, 7, 8, 11, 12, 17, 18, 21, 22, 29, 30, 33, 34, 39, 40, 43, 44, 9, 10, 13, 14, 19, 20, 23, 24, 31, 32, 35, 36, 41, 42, 45, 46, 55, 56, 59, 60, 65, 66, 69, 70, 77, 78, 81, 82, 87, 88, 91, 92, 11, 12, 15, 16, 21, 22, 25, 26, 33, 34, 37, 38, 43, 44, 47, 48, 57, 58, 61, 62

Offset: 1

Author

Hermann Gruber, Feb 12 2022

Keywords

• nonn
• tabf

Comments

Following the notation in Gruber/Holzer (2021), for n >= 1 and 1 <= k <= 2^n, let P'{n,k} denote the set containing the lexicographically first k palindromes of odd length 2n-1 over the binary alphabet {a,b}. T(n,k) is the minimum number of alphabetic symbols in any regular expression describing the set P'{n,k}.

Examples

			Triangle T(n,k) begins:
  1, 2;
  3, 4,  7,  8;
  5, 6,  9, 10, 15, 16, 19, 20;
  7, 8, 11, 12, 17, 18, 21, 22, 29, 30, 33, 34, 39, 40, 43, 44;
  ...
		

Links

• Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length, Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science, Article No. 53, pp. 53:1-53:15, 2021.

Crossrefs

Cf. A351489 gives the corresponding irregular triangle for even length 2*n.

Programs

Mathematica

Flatten[Table[2n+3(k-1)-2Total[IntegerDigits[k-1,2]]-1,{n,6},{k,2^n}]] (* Stefano Spezia, Feb 13 2022 *)

PARI

T(n,k) = 2*n + 3*(k-1) - 2*hammingweight(k-1) - 1 \\ Andrew Howroyd, Feb 12 2022

Formula

T(n,k) = 2*n + 3*(k-1) - 2*hamming_weight(k-1)-1. See theorem 20 in Gruber/Holzer (2021).