cp's OEIS Frontend

Sequence S = UABV does not satisfy the desired property if nonempty blocks A and B have the same average (where U, V, or both may be empty). For example, 4,1,3,5,0,6,2,9 does not have the desired property, because it can be written as (4)(1,3,5)(0,6)(2,9) and the two consecutive blocks (1,3,5) and (0,6) have the same average 3.

The Gerver-Ramsey theorem implies that for each n, such a sequence is of bounded length; see Theorem 2 in the paper of Brown.

For all n <= 9 except n = 7 and 8, there exists a longest sequence that is also palindromic. - Pontus von Brömssen, Jan 09 2025

a(n) is the last element of row n in A379998.

The sequences counted by a(n) have length A379914(n).

a(n) is even for all n >= 2, because each term x in a sequence can be replaced by n-1-x, giving another sequence of maximum length.

See A379914 for details.

a(n) is even if and only if n is odd; for each sequence can be paired up with its complement (where all terms x are replaced by n-1-x), and the only self-complementary sequences (that also satisfy the consecutive blocks condition) are the empty sequence and those with a single term (n-1)/2 for odd n.