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A383192 a(n) is the number of possible choices for the first n terms of a "mean-central" sequence, where a monotonically increasing sequence of positive integers {b(n)} is called "mean-central" if for each positive integer k, the arithmetic mean of the first b(k) terms is exactly b(k).

%I A383192 #27 Apr 29 2025 13:16:59
%S A383192 1,2,2,3,3,4,8,16,20,25,27,48,72,107,149,260,372,511,653,1032,1192,
%T A383192 1713,2218,3992,5504,7729,10452,16397,21700,32292,43742,72859,98926,
%U A383192 143759,187703,284689,368374,526256,729299,1315303
%N A383192 a(n) is the number of possible choices for the first n terms of a "mean-central" sequence, where a monotonically increasing sequence of positive integers {b(n)} is called "mean-central" if for each positive integer k, the arithmetic mean of the first b(k) terms is exactly b(k).
%C A383192 Suppose that the initial terms (b(1), ..., b(n)) are chosen such that for each b(k) <= n, the arithmetic mean of the first b(k) terms is exactly b(k). Then the following terms b(n+1), ..., b(b(n)) can be split into parts, the sums of which are fixed. Greedily choose the terms so that for each part, the last term is as small as possible. If the monotonicity still cannot be satisfied, the initial terms are invalid. Otherwise, we can fill the remaining terms with b(k) = 2*k - 1, forming a mean-central sequence.
%H A383192 Art of Problem Solving, <a href="https://artofproblemsolving.com/community/c6h3549401p34560823">European Girls' Mathematical Olympiad 2025 Problem 2</a>
%H A383192 Yifan Xie, <a href="/A383192/a383192_1.txt">Python program</a>
%e A383192 For n = 4, the 3 valid choices for the first 4 terms of a central sequence are (1, 3, 5, 6), (1, 3, 5, 7) and (1, 4, 5, 6).
%e A383192 (1, 3, 5, 6, 10, 11, 13, 15, ...), (1, 3, 5, 7, 9, ...) and (1, 4, 5, 6, 9, 11, 13, ...) are the corresponding continuations.
%e A383192 Although the initial terms meet the requirement, (1, 3, 5, 8) is invalid because for the arithmetic mean of the first 5 terms to be 5, b(5) must be 8, breaking the monotonicity.
%Y A383192 Cf. A383193, A383194.
%K A383192 nonn,more
%O A383192 1,2
%A A383192 _Yifan Xie_, Apr 19 2025
%E A383192 Definition edited by _N. J. A. Sloane_, Apr 29 2025