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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.

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%I A331859 #23 Feb 12 2020 01:08:04
%S A331859 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,
%T A331859 16,16,17,17,17,17,18,18,18,19,19,19,19,20,20,20,20,20,21,21,21,21,22,
%U A331859 22,22,22,23,23,23,23,23,24,24,24,24,24,25,25,25,25,25
%N A331859 The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.
%C A331859 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.)
%C A331859 a(100^n) = A011545(n).
%C A331859 Since arctan(sqrt(1/n)) is approximately sqrt(1/n) for large values of n, a(n) = A121854(n) for most values of n.
%C A331859 Conjecture: The values of n for which a(n) != A121854(n) is a subset of A331903.
%C A331859 Initial phase:
%C A331859    \ |                            ______________________
%C A331859   \ \|                           |                      |
%C A331859    \ |                           |                      |
%C A331859   \ \|                           |                      |
%C A331859    \ |                           |                      |
%C A331859   \ \|                      <=== |       Block A        |
%C A331859    \ |         _________         |                      |
%C A331859   \ \|        |         |        |        M = n         |
%C A331859    \ |        | Block B |        |                      |
%C A331859   \ \|        |    |    |        |                      |
%C A331859    \ |        |  M = 1  |        |                      |
%C A331859   \ \|        |_________|        |______________________|
%C A331859    \ L----------------------------------------------------------
%C A331859   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
%C A331859   \ \|
%C A331859    \ |                    ______________________
%C A331859   \ \|                   |                      |
%C A331859    \ |                   |                      |
%C A331859   \ \|                   |                      |
%C A331859    \ |                   |                      |
%C A331859   \ \|              <=== |                      |
%C A331859    \ |         _________ |                      |
%C A331859   \ \|        |         ||                      |
%C A331859    \ |        |         ||                      |
%C A331859   \ \|        |         ||                      |
%C A331859    \ |        |         ||                      |
%C A331859   \ \|        |_________||______________________|
%C A331859    \ L----------------------------------------------------------
%C A331859   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
%C A331859   \ \|
%C A331859    \ |                   ______________________
%C A331859   \ \|                  |                      |
%C A331859    \ |                  |                      |
%C A331859   \ \|                  |                      |
%C A331859    \ |                  |                      |
%C A331859   \ \|              <== |                      |
%C A331859    \ |      _________   |                      |
%C A331859   \ \|     |         |  |                      |
%C A331859    \ |     |         |  |                      |
%C A331859   \ \|<===>|         |  |                      |
%C A331859    \ |     |         |  |                      |
%C A331859   \ \|     |_________|  |______________________|
%C A331859    \ L----------------------------------------------------------
%C A331859   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
%H A331859 Peter Kagey, <a href="/A331859/b331859.txt">Table of n, a(n) for n = 1..10000</a>
%H A331859 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/198615/53884">Elastic collisions between blocks</a>
%H A331859 Grant Sanderson, <a href="https://www.quantamagazine.org/how-pi-connects-colliding-blocks-to-a-quantum-search-algorithm-20200121/">How Pi Connects Colliding Blocks to a Quantum Search Algorithm</a>, Quanta Magazine (2020).
%H A331859 Grant Sanderson, <a href="https://www.youtube.com/watch?v=HEfHFsfGXjs">The most unexpected answer to a counting puzzle</a>, 3Blue1Brown video (2019)
%H A331859 Grant Sanderson, <a href="https://www.youtube.com/watch?v=jsYwFizhncE">Why do colliding blocks compute pi?</a>, 3Blue1Brown video (2019)
%F A331859 a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1.
%t A331859 Table[Ceiling[Pi/ArcTan[Sqrt[1/n]] - 1], {n, 1, 100}]
%Y A331859 Cf. A121854, A331903, A331904.
%K A331859 nonn
%O A331859 1,1
%A A331859 _Peter Kagey_, Jan 29 2020