Descending Dungeons and Iterated Base-Changing

For real numbers a, b> 1, let as a_b denote the result of interpreting a in base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to be numbers of the form a_b_c_d_..._e, parenthesized either from the bottom upwards (preferred) or from the top downwards. Among other things, we show that the sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or n_(n-1)_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the logarithms are to the base 10. We also investigate the behavior as n increases of the sequence a_a_a_..._a, with n a's, parenthesized from the bottom upwards. This converges either to a single number (e.g. to the golden ratio if a = 1.1), to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a = frac{100{99).Comment: 11 pages; new version takes into account comments from referees; version of Sep 25 2007 inculdes a new theorem and several small improvement

Seven Staggering Sequences

When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column that "every recreational mathematician should buy a copy forthwith." That book contained 2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains 117000 sequences. This paper will describe seven that I find especially interesting. These are the EKG sequence, Gijswijt's sequence, a numerical analog of Aronson's sequence, approximate squaring, the integrality of n-th roots of generating functions, dissections, and the kissing number problem. (Paper for conference in honor of Martin Gardner's 91st birthday.)Comment: 12 pages. A somewhat different version appeared in "Homage to a Pied Puzzler", E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-11

2178 And All That

For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood.Comment: 22 pages, 16 figures, one table. July 4 2013: corrected typo in table, added conjectures about particular graphs. Sept. 24 2013: corrected typos, added conjectures and theorems. Oct 13 2013: minor edit