Автор |
Cassaigne, Julien |
Автор |
Finch, Steven R. |
Дата выпуска |
1995 |
dc.description |
For odd v ≥ 5, Schmerl and Spiegel have proved that the 1-additive sequence (2, v) has precisely two even terms and, consequently, is regular. For 5 ≤ v ≡ 1 mod 4, we prove, using a different approach, that the 1-additive sequence (4, v) has precisely three even terms. The proof draws upon the periodicity properties of a certain ternary quadratic recurrence.Unlike the case of (2, v), the regularity of (4, v) can be captured by expressions in closed form. For example, its period can be written as an exponential sum of binary digit sums. Therefore the asymptotic density δ(v) of (4, v) tends to 0 as v → ∞, but is misbehaved in the sense thatThis is proved using techniques adapted from Harborth and Stolarsky. |
Формат |
application.pdf |
Издатель |
Taylor & Francis Group |
Копирайт |
Copyright Taylor and Francis Group, LLC |
Название |
A Class of 1-Additive Sequences and Quadratic Recurrences |
Тип |
research-article |
DOI |
10.1080/10586458.1995.10504307 |
Electronic ISSN |
1944-950X |
Print ISSN |
1058-6458 |
Журнал |
Experimental Mathematics |
Том |
4 |
Первая страница |
49 |
Последняя страница |
60 |
Аффилиация |
Cassaigne, Julien; LITP, Institut Blaise Pascal |
Аффилиация |
Finch, Steven R.; <sup>b</sup> 6 Foster Street, Wakefield, MA, 01880, USA E-mail: sfinch@gnu.aLmit.edu |
Выпуск |
1 |
Библиографическая ссылка |
Finch, S. R. 1991. “Conjectures about -additive sequences”. Fibonacci Quart., 29: 209–214. [Finch 1991] |
Библиографическая ссылка |
Finch, S. R. 1992. “On the regularity of certain 1-additive sequences”. J. Combin. Theory, A60: 123–130. [Finch 1992a] |
Библиографическая ссылка |
Finch, S. R. 1992. “Patterns in 1-additive sequences”. Experimental Math., : 57–63. [Finch 1992b] |
Библиографическая ссылка |
Harborth, H. 1977. “Number of odd binomial coefficients”. Proc. Amer. Math. Soc., 62: 19–22. [Harborth 1977] |
Библиографическая ссылка |
Long, C. T. 1981. “Pascal's triangle modulo p'. Fibonacci Quart., 19: 458–463. [Long 1981] |
Библиографическая ссылка |
Queneau, R. 1972. “Sur les suites -additives”. J. Combin. Theory, A12: 31–71. [Queneau 1972], English summary in Math. Rev. 46, 1741 |
Библиографическая ссылка |
Schmerl, J. H. and Spiegel, E. 1994. “The regularity of some 1-additive sequences”. J. Comb. Theory, A66: 172–175. [Schmerl and Spiegel 1994] |
Библиографическая ссылка |
Stolarsky, K. B. 1977. “Power and exponential sums of digital sums related to binomial coefficient parity”. SIAM J. Appl. Math., 32: 717–730. [Stolarsky 1977] |
Библиографическая ссылка |
Ulam, S. M. 1964. Problems in Modern Mathematics New York: Interscience.. [Ulam 1964] |
Библиографическая ссылка |
Wolfram, S. 1984. “Geometry of binomial coefficients”. Amer. Math. Monthly, 91: 566–571. [Wolfram 1984] |