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A351490
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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.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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}.
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LINKS
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FORMULA
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T(n,k) = 2*n + 3*(k-1) - 2*hamming_weight(k-1)-1. See theorem 20 in Gruber/Holzer (2021).
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EXAMPLE
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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;
...
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MATHEMATICA
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Flatten[Table[2n+3(k-1)-2Total[IntegerDigits[k-1, 2]]-1, {n, 6}, {k, 2^n}]] (* Stefano Spezia, Feb 13 2022 *)
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PROG
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(PARI) T(n, k) = 2*n + 3*(k-1) - 2*hammingweight(k-1) - 1 \\ Andrew Howroyd, Feb 12 2022
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CROSSREFS
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Cf. A351489 gives the corresponding irregular triangle for even length 2*n.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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