This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379914 #28 Jan 09 2025 12:18:34
%S A379914 1,3,3,7,9,19,20,31,37
%N A379914 Length of longest sequence over {0,1,...,n-1} containing no two consecutive blocks with the same average.
%C A379914 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.
%C A379914 The Gerver-Ramsey theorem implies that for each n, such a sequence is of bounded length; see Theorem 2 in the paper of Brown.
%C A379914 For all n <= 9 except n = 7 and 8, there exists a longest sequence that is also palindromic. - _Pontus von Brömssen_, Jan 09 2025
%H A379914 Tom Brown, <a href="https://math.colgate.edu/~integers/m22/m22.pdf">Approximations of additive squares in infinite words</a>, Integers 12 (2012), #A22.
%e A379914 For 1 <= n <= 9, the lexicographically least sequences achieving the given bound are as follows:
%e A379914 n=1: 0
%e A379914 n=2: 010
%e A379914 n=3: 010
%e A379914 n=4: 0203202
%e A379914 n=5: 010343010
%e A379914 n=6: 0501050254520501050
%e A379914 n=7: 03143656151050356353
%e A379914 n=8: 1250673747530401046047606760502
%e A379914 n=9: 0323725782750730106010370572875273230
%Y A379914 Cf. A379998, A379999, A380000.
%K A379914 nonn,more
%O A379914 1,2
%A A379914 _Jeffrey Shallit_, Jan 06 2025
%E A379914 a(9) from _Pontus von Brömssen_, Jan 07 2025