A351491 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 length 2*n over a ternary alphabet, n >= 0, 1 <= k <= 3^n.

0, 2, 4, 6, 4, 6, 8, 12, 14, 16, 20, 22, 24, 6, 8, 10, 14, 16, 18, 22, 24, 26, 32, 34, 36, 40, 42, 44, 48, 50, 52, 58, 60, 62, 66, 68, 70, 74, 76, 78, 8, 10, 12, 16, 18, 20, 24, 26, 28, 34, 36, 38, 42, 44, 46, 50, 52, 54, 60, 62, 64, 68, 70, 72, 76, 78, 80, 88

Offset: 0

Author

Hermann Gruber, Feb 13 2022

Keywords

• nonn
• easy
• tabf

Comments

Analogous to A351489 (which is the corresponding sequence for palindromes over binary alphabet).

Examples

			Triangle T(n,k) begins:
      k=1  2   3   4   5   6 ...
  n=0:  0,
  n=1:  2, 4,  6;
  n=2:  4, 6,  8, 12, 14, 16, 20, 22, 24;
  n=3:  6, 8, 10, 14, 16, 20, 22, 24, 26, 32, 34, 36, 40, 42, 44, 48, 50, 52, 58, 60, 62, 66, 68, 70, 74, 76, 78;
  ...
		

References

• Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length. Extended journal version, in preparation, 2022.

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. A053735 (ternary sum of digits), A351489 (for binary alphabet).

Formula

Let SumOfDigitsInBase(m,b) denote the digit sum of nonnegative integer m in base b. Then the general formula for alphabet size q reads as

T(n,k) = 2*n + (2*q*(k-1))/(q-1) - (2*SumOfDigitsInBase(k-1,q))/(q-1). [Gruber and Holzer 2022 theorem 27]