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

1,2

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.

Links

Table of n, a(n) for n=1..40.

Art of Problem Solving, European Girls' Mathematical Olympiad 2025 Problem 2

Yifan Xie, Python program

Example

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.

Crossrefs

Cf. A383193, A383194.

Sequence in context: A283363 A236210 A357416 * A362251 A165120 A165129

Adjacent sequences: A383189 A383190 A383191 * A383193 A383194 A383195

Keyword

nonn,more

Author

Yifan Xie, Apr 19 2025

Extensions

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

Status

approved