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A331859
The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.
4
3, 5, 5, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25
OFFSET
1,1
COMMENTS
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. (See Grant Sanderson links for animated examples.)
a(100^n) = A011545(n).
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.
Initial phase:
\ | ______________________
\ \| | |
\ | | |
\ \| | |
\ | | |
\ \| <=== | Block A |
\ | _________ | |
\ \| | | | M = n |
\ | | Block B | | |
\ \| | | | | |
\ | | M = 1 | | |
\ \| |_________| |______________________|
\ L----------------------------------------------------------
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \|
\ | ______________________
\ \| | |
\ | | |
\ \| | |
\ | | |
\ \| <=== | |
\ | _________ | |
\ \| | || |
\ | | || |
\ \| | || |
\ | | || |
\ \| |_________||______________________|
\ L----------------------------------------------------------
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \|
\ | ______________________
\ \| | |
\ | | |
\ \| | |
\ | | |
\ \| <== | |
\ | _________ | |
\ \| | | | |
\ | | | | |
\ \|<===>| | | |
\ | | | | |
\ \| |_________| |______________________|
\ L----------------------------------------------------------
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
LINKS
Code Golf Stack Exchange, Elastic collisions between blocks
Grant Sanderson, How Pi Connects Colliding Blocks to a Quantum Search Algorithm, Quanta Magazine (2020).
Grant Sanderson, The most unexpected answer to a counting puzzle, 3Blue1Brown video (2019)
Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019)
FORMULA
a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1.
MATHEMATICA
Table[Ceiling[Pi/ArcTan[Sqrt[1/n]] - 1], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Kagey, Jan 29 2020
STATUS
approved