OFFSET
0,3
COMMENTS
As shown in the proof of [Gruber and Holzer, lemma 9], the maximum is attained by choosing i as the largest number not exceeding n whose ternary representation is (33...3)_4. Also by lemma 6, for this choice of i we have A053737(i) = 3*floor(log_4(n+1)) and A053737(n-i) = A053737(n+1)-1, giving the formula below.
REFERENCES
Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length. Extended journal version, in preparation, 2024.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10000
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
a(n) = 3*floor(log_4(n+1)) + A053737(n+1) - 1 [Gruber and Holzer, lemma 9].
EXAMPLE
For n=74, the maximum is attained by 63 + 11 = (333)_4 + (23)_4. Using 75=(1023)_4, comparing with the formula above, A053737(63) = 3*floor(log_4(n+1)) = 9 and A053737(11) = A053737(74+1)-1 = 5. Notice that other pairs attain the maximum as well. Namely, 43 + 31 = (223)_4 + (133)_4, as well as 47 + 27 = (233)_4 + (123)_4, and 59 + 15 = (323)_4 + (33)_4.
MATHEMATICA
Table[3*Floor[Log[4, k]] + DigitSum[k, 4] - 1, {k, 100}] (* Paolo Xausa, Aug 01 2024 *)
PROG
(PARI) a(n) = 3*logint(n+1, 4) + sumdigits(n+1, 4) - 1;
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Hermann Gruber, Jun 26 2024
STATUS
approved