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a(1)=a(2)=1, a(n+2) = a(n+1)+1 if sign(sin(a(n+1))) = sign(sin(a(n))), a(n+2) = a(n)+1 otherwise.
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%I #15 Jan 01 2024 13:59:35

%S 1,1,2,3,4,4,5,6,7,7,8,9,10,10,11,12,13,13,14,15,16,16,17,18,19,19,20,

%T 21,22,22,23,24,25,26,26,27,28,29,29,30,31,32,32,33,34,35,35,36,37,38,

%U 38,39,40,41,41,42,43,44,44,45,46,47,48,48,49,50,51,51,52,53,54,54,55

%N a(1)=a(2)=1, a(n+2) = a(n+1)+1 if sign(sin(a(n+1))) = sign(sin(a(n))), a(n+2) = a(n)+1 otherwise.

%C Asymptotically a(n)~C*n with C=0.75854... Is this constant rational?

%C The constant C=0.75854... is not rational. Otherwise it would imply the rationality of Pi since C = Pi/(Pi+1) = 0.7585469930... For n >= 6 we have a(n) = (n-1)-k with k >= 1 such that ceiling(k*(Pi+1)) <= (n-1) < ceiling((k+1)*(Pi+1)) = ceiling(k*(Pi+1)) + 4 or 5. So n-1 = k*(Pi+1) + O(1) and then: a(n) = (Pi/(Pi + 1))*n + O(1). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 15 2002

%K easy,nonn

%O 1,3

%A _Benoit Cloitre_, Apr 11 2002