Record-setters in the Stern sequence. (English) Zbl 1533.11064
Summary: Stern’s diatomic series, denoted by \((a (n))_{n \geq 0}\), is defined by the recurrence relations \(a(2 n) = a(n)\) and \(a(2 n + 1) = a(n) + a(n + 1)\) for \(n \geq 1\), with initial values \(a(0) = 0\) and \(a(1) = 1\). A record-setter for a sequence \((s (n))_{n \geq 0}\) is an index \(v\) such that \(s(i) < s(v)\) holds for all \(i < v\). In this paper, we provide a complete description of the record-setters for the Stern sequence. As a consequence, we prove that the running max sequence of the Stern sequence is not 2-regular.
MSC:
| 11B83 | Special sequences and polynomials |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 05A16 | Asymptotic enumeration |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Numbers n such that fusc(k) < fusc(n) for all k < n, where fusc is Stern’s diatomic series.Record values in A002487.
References:
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