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%I #46 Jul 05 2024 11:11:19
%S 1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,
%T 1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,
%U 0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1
%N a(n) = floor(n*Pi) mod 2.
%C The arithmetic mean (1/(n+1))*Sum_{k=0...n} a(k) converges to 1/2. What is effectively the same: the Cesaro limit (C1) of a(n) is 1/2. When we pick a term of the sequence at random, the probability of getting a '1' is 1/2. If we select a '1' randomly, the probability p11 of finding a '1' as the next term right of it is p11 = Pi - 3. If we select a '1' randomly, the probability p10 of finding a '0' as the next term right of it is p10 = 4 - Pi. Analogous statements hold for '0' --> '0' (p00 = p11) and '0' --> '1' (p01 = p10).
%C First differs from A195062 at a(113). - _Alois P. Heinz_, Jan 22 2012
%H Paolo Xausa, <a href="/A115788/b115788.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Borwein and Loki Jörgenson, <a href="https://doi.org/10.1080/00029890.2001.11919824">Visible Structures in Number Theory</a>, The American Mathematical Monthly, Vol. 108, 2001, No. 10, pp. 906-908.
%F a(n) = floor(n*Pi) mod 2.
%F a(n) = A000035(A022844(n)). - _Michel Marcus_, Jul 05 2024
%e a(2) = 0 because floor(2*Pi) = floor(6.28... ) = 6,
%e a(8) = 1 because floor(8*Pi) = floor(25.13...) = 25.
%p a:= proc(n) Digits:= length(n) +15; floor(n*Pi) mod 2 end:
%p seq(a(n), n=1..150); # _Alois P. Heinz_, Jan 22 2012
%t Floor[Mod[\[Pi] Range[110],2]] (* _Harvey P. Dale_, Apr 02 2011 *)
%Y Cf. A000035, A022844, A063438, A115789, A115790, A372544.
%Y Cf. A195062.
%K nonn
%O 1
%A _Hieronymus Fischer_, Jan 31 2006
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