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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.
MSC:
 11B50 Sequences (mod $$m$$) 11B65 Binomial coefficients; factorials; $$q$$-identities 05A10 Factorials, binomial coefficients, combinatorial functions
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