This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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