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On a conjecture of McIntosh regarding LP-sequences. (English) Zbl 1165.11021
An LP sequence is a sequence \((a_n)_{n\geq 0}\) over \(\mathbb{Z}\), such that for any prime \(p\) and any natural integer \(n\), if \(n= \sum n_ip^i\) is the base \(p\) expansion of \(n\), then \(a_n\equiv\prod a_{n_i}\pmod p\). The conjecture of R. J. McIntosh [Am. Math. Mon. 99, No. 3, 231–238 (1992; Zbl 0755.11001)] reads: if \((a_n)\) is a nonnegative LP sequence with \(a_n= O(b^n)\) for some \(b< e\), then \((a_n)\) is one of the four sequences \((000\dots)\), \((100\dots)\), \((111\dots)\), \((1248\dots)\).
The author of the paper under review formulates a modified conjecture after proving that the sequence \((1\,0{2\choose 1}\,0{4\choose 2}\,0{6\choose 3}\dots)\) obtained from the middle binomial coefficients should be added to the four sequences above. Furthermore, he proves that the modified conjecture is true if the generating function \(\sum a_n x^n\) is supposed to be algebraic over \(\mathbb{Q}(x)\). Finally, the set of LP sequence \((a_n)_{n\geq 0}\) with \(a_n= O(b^n)\) for some \(b< e\) is proved to be countable, while for any \(b\geq e\) the set of LP sequences \((a_n)_{n\geq 0}\) with \(\lim(a_n)^{1/n}= b\) is proved to have the power of continuum.
11B50 Sequences (mod \(m\))
11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
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