A351489 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 binary alphabet, n >= 0, 1 <= k <= 2^n.

0, 2, 4, 4, 6, 10, 12, 6, 8, 12, 14, 20, 22, 26, 28, 8, 10, 14, 16, 22, 24, 28, 30, 38, 40, 44, 46, 52, 54, 58, 60, 10, 12, 16, 18, 24, 26, 30, 32, 40, 42, 46, 48, 54, 56, 60, 62, 72, 74, 78, 80, 86, 88, 92, 94, 102, 104, 108, 110, 116, 118, 122, 124, 12, 14, 18, 20, 26, 28, 32, 34, 42, 44, 48, 50, 56, 58, 62

Offset

0,2

Comments

Following the notation in Gruber/Holzer (2021), for n >= 0 and 1 <= k <= 2^n, let P_{n,k} denote the set containing the lexicographically first k palindromes of even length 2n 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}.

Links

Table of n, a(n) for n=0..77.

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.

Formula

T(n,k) = 2*n + 4*(k-1) - 2*wt(k-1), where wt(n) = A000120(n) is the sum of the binary digits of n. [Gruber and Holzer theorem 14]

Example

Triangle T(n,k) begins:
      k=1   2   3   4   5   6 ...
  n=0:  0,
  n=1:  2,  4;
  n=2:  4,  6, 10, 12;
  n=3:  6,  8, 12, 14, 20, 22, 26, 28;
  n=4:  8, 10, 14, 16, 22, 24, 28, 30, 38, 40, 44, 46, 52, 54, 58, 60;
  ...

Mathematica

Flatten[Table[2n+4(k-1)-2Total[IntegerDigits[k-1, 2]], {n, 0, 6}, {k, 2^n}]] (* Stefano Spezia, Feb 13 2022 *)

Crossrefs

Cf. A000120 (sum of binary digits), A351490 (on odd lengths).

Sequence in context: A237828 A362607 A340626 * A022471 A359294 A224487

Adjacent sequences: A351486 A351487 A351488 * A351490 A351491 A351492

Keyword

nonn,easy,tabf

Author

Hermann Gruber, Feb 12 2022

Status

approved