A165465 Positions of zeros in A165464. Fixed points of A166041/A166042.

0, 1, 7, 8, 15, 16, 22, 23, 24, 25, 1702855, 1702856, 1702857, 1702872, 1702873, 2220150, 3327583, 3329174, 3329270, 3329271, 3329279

Offset: 0

Author

Antti Karttunen, Oct 06 2009

Keywords

• nonn
• more

Comments

Consider two immortal sage kings traveling on the infinite chessboard, visiting every square at the 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 the day zero (being sages, they can comfortably stay in the same square without bloodshed). One decides to follow the Hilbert curve on his never-ending journey, while the other follows the Peano curve. (These are both illustrated in the entry A166041.) This sequence gives the days when they will meet, when they both come to the same square on the same day.

Both walk first one square towards east, where they meet at Day 1. Then one turns south, while the other one proceeds to the east. However, just six days later, on Day 7, they meet again, at the square (2,1), two squares south and one square east of the starting corner. They also meet the next day (Day 8), as well as another week later (Day 15), and before January is over, they meet still five more times, on Days 16, 22, 23, 24 and 25. However, it takes 4662 years and about three months before they meet again, on three successive days (Days 1702855, 1702856 and 1702857). - Antti Karttunen, Oct 13 2009 [Edited to Hilbert vs Peano by Kevin Ryde, Aug 30 2020]

Subset of A165480. - Antti Karttunen, Oct 13 2009

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20

Crossrefs

Cf. A165467, A165480, A163901.