A331904 -id:A331904 - cp's OEIS frontend

A331859 The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.

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

Peter Kagey, Jan 29 2020

nonn

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----------------------------------------------------------
  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Peter Kagey, Table of n, a(n) for n = 1..10000
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)

Cf. A121854, A331903, A331904.

Table[Ceiling[Pi/ArcTan[Sqrt[1/n]] - 1], {n, 1, 100}]

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

A331903 Positions of records in A331859.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 26, 29, 33, 36, 40, 45, 49, 53, 58, 63, 68, 74, 79, 85, 91, 97, 104, 110, 117, 124, 131, 139, 146, 154, 162, 170, 179, 187, 196, 205, 214, 224, 233, 243, 253, 263, 274, 284, 295, 306, 318, 329, 341, 353, 365, 377, 389

Offset: 1

Peter Kagey, Jan 31 2020

nonn

Peter Kagey, Table of n, a(n) for n = 1..2000

Cf. A331859, A331904.

cp's OEIS Frontend

A331859 The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.

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