the point at coordinates (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24) in 24-dimensional space is exactly 70 units from the origin, because the sum of the first 24 squares is 4900. (this amazing relation is somehow related to the 24-dimensional Leech lattice, a different lattice than our hypercube lattice of all points with integer coordinates.) the point is one of 905754973269671819419476064818886581168 lattice points 70 units from the origin. that large number divided by 24 factorial is approximately 1459839320625890.4 : there are many more lattice points on a hypersphere of radius 70 than just the permutations of the coordinates of our special point.
our special point is one of 370123270431636382177726611489013547080401 lattice points inside or on the boundary of a 24-dimensional ball of radius 70. this volume is approximately 408.6 times larger than the above surface "area" of lattice points. in other words, this outermost layer of lattice points does not dominate the total lattice points. this matches lower dimensional intuition, which is rare when considering volumes in high dimensions.
future work: actual volume, surface area, and ratio of a 24-dimensional sphere of radius 70. this should be easy.
the above results are derived from the b-file of OEIS A000156. I confirmed the list up to entry 1000 (with 48 hours of computing). I do not know how its larger entries (up to entry 10000) were calculated; there must be some algorithmic tricks beyond what I implemented.
I discovered A000156 by calculating its first few terms (future post kiqanzug), then searching OEIS for the continuation. this a standard way to use the database.
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