bibkey: kagey2020a331859 authors: Peter Kagey year: 2020 title: “OEIS A331859, The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall” doi: null url: https://oeis.org/A331859 claim: “The entry counts the elastic collisions of the colliding-blocks system with masses n and 1 and a wall, gives the formula a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1, and conjectures that the n with a(n) != floor(Pi*sqrt(n)) are positions of records of the sequence.” strata_touched:

• D5/S3/Constants/Billiards/CollidingBlocksRecords license: citation-only triage: anchor

OEIS A331859

The COMMENTS of A331859 describe the system:

Suppose there is a block A of mass n sliding left toward a stationary block B of mass 1, to the left of which is a wall. Assuming the sliding is frictionless and the collisions are elastic, a(n) is the number of collisions between A and B plus the number of collisions between B and the wall.

and continue

Since arctan(sqrt(1/n)) is approximately sqrt(1/n) for large values of n, a(n) = A121854(n) for most values of n.

Conjecture: The values of n for which a(n) != A121854(n) is a subset of A331903.

The FORMULA section gives

a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1.

the closed form of the collision count; its identification with the collision dynamics is attributed to G. Galperin (2003), whose paper was not read for this note. A121854 is a(n) = floor(Pi*(sqrt(n))) and A331903 is Positions of records in A331859. The entry links the 3Blue1Brown videos by Grant Sanderson on colliding blocks and Quanta Magazine (2020).

Verified locator

• URL: https://oeis.org/A331859
• Version and location: OEIS A331859, revision 27 (2025-09-18), COMMENTS and FORMULA; entry by Peter Kagey, 2020-01-29.
• Related entries: https://oeis.org/A121854 (revision 22), https://oeis.org/A331903 (revision 12).