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%I #31 Jan 27 2022 08:01:59
%S 3,3,4,4,5,5,6,6,6,7,7,7,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,11,12,
%T 12,12,12,12,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,15,16,
%U 16,16,16,16,16,17,17,17,17,17,17,17,18,18,18,18,18,18,18,18,19,19,19,19
%N a(n) = floor(Pi/arccos(n/(n+1))).
%C Let N(m1, m2, m3) be the maximum possible number of pairwise elastic collisions in a dynamic system of 3 point masses m1, m2, m3 on a line. N(m1,m2, m3) is independent of initial velocities and coordinates of masses m1, m2, m3. If m1 = n*m2 = m3 then N(m1, m2, m3) = -[-Pi/arccos(sqrt(m1*m3/((m1+m2)*(m2+m3))))] = a(n).
%H Muniru A Asiru, <a href="/A119661/b119661.txt">Table of n, a(n) for n = 1..1000</a>
%H G. A. Galperin, A. N. Zemliakov, <a href="https://math.ru/lib/book/djvu/bib-kvant-15/Kv77-90_Matematicheskie_Bilyardi_G.A.Galperin.djvu">Mathematical Billiards</a>, "KVANT" Library, Issue 77, Moscow, Nauka, 1990, p. 165. (in Russian)
%e n = 24, -Pi/arccos(n/(n+1)) = -11.06997134, [-11.06997134] = -11. Therefore a(24) = 11.
%p seq(-trunc(-Pi/arccos(n/(n+1))), n=1..76); # _Peter Luschny_, Jun 28 2018
%t Table[ -IntegerPart[ -Pi/ArcCos[ n/(n+1) ] ], {n, 1, 100}]
%K nonn
%O 1,1
%A _Alexander Adamchuk_, Jul 28 2006
%E Edited by _Peter Luschny_, Jun 29 2018
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