OFFSET
0,2
COMMENTS
Here is a little parable for illustrating the magnitudes of the numbers involved. Consider two immortal sage kings traveling on the infinite chessboard, visiting every square at a leisurely pace of one square per day. Both start their journey at the beginning of the year from the upper left-hand corner square at Day 0 (being sages, they can comfortably stay in the same square). One decides to follow the Hilbert curve (as in A163357) on his never-ending journey, while the other follows the Peano curve (as in A163336; both walks are illustrated in entry A166043). This sequence gives the days when they will meet, when they both arrive at the same square on the same day.
From the corner, one king walks first towards the east, while the other walks towards the south, so their paths diverge at the beginning. However, about a week later (Day 8), they meet again on square (2,2), two squares south and two squares east of the starting corner. The next day they are both traveling towards the south, so they meet also on Day 9, at square (3,2). After that, they meet briefly three months later (Day 105), and also about three years later (Day 1126), after which they loathe each other so much that they both walk in solitude for the next 18189 (eighteen thousand one hundred and eighty nine) years before they meet again, total of eleven times in just about one month's time (days 6643718-6643752). - Antti Karttunen, Oct 13 2009 [Edited to Hilbert vs Peano by Kevin Ryde, Aug 29 2020]
LINKS
A. Karttunen, Table of n, a(n) for n = 0..29
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 06 2009
STATUS
approved