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