A374056 a(n) = max_{i=0..n} S_4(i) + S_4(n-i) where S_4(x) = A053737(x) is the base-4 digit sum of x.

0, 1, 2, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14

Offset: 0

Author

Hermann Gruber, Jun 26 2024

Keywords

• nonn
• easy
• base

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.

Examples

			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.
		

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.

Crossrefs

Cf. A053737, A102572, A014701, A374054.

Programs

Mathematica

Table[3*Floor[Log[4, k]] + DigitSum[k, 4] - 1, {k, 100}] (* Paolo Xausa, Aug 01 2024 *)

PARI

a(n) = 3*logint(n+1, 4) + sumdigits(n+1, 4) - 1;

Formula

a(n) = 3*floor(log_4(n+1)) + A053737(n+1) - 1 [Gruber and Holzer, lemma 9].