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).

1, 2, 2, 3, 3, 4, 8, 16, 20, 25, 27, 48, 72, 107, 149, 260, 372, 511, 653, 1032, 1192, 1713, 2218, 3992, 5504, 7729, 10452, 16397, 21700, 32292, 43742, 72859, 98926, 143759, 187703, 284689, 368374, 526256, 729299, 1315303

Offset: 1

Author

Yifan Xie, Apr 19 2025

Keywords

• nonn
• more

Comments

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.

Examples

			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).
(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.
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.
		

Links

• Art of Problem Solving, European Girls' Mathematical Olympiad 2025 Problem 2
• Yifan Xie, Python program

Crossrefs

Cf. A383193, A383194.

Extensions

Definition edited by N. J. A. Sloane, Apr 29 2025