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%I #22 Feb 23 2024 10:47:05
%S 1,0,-1,2,-3,3,-2,1,0,0,-1,1,-1,2,-2,0,2,-2,1,0,-1,1,0,-1,1,0,0,-1,2,
%T -3,3,-2,1,0,-1,1,0,-1,2,-2,0,2,-2,1,-1,2,-2,0,2,-2,1,-1,1,0,0,0,-1,1,
%U -1,2,-2,0,2,-2,1,-1,2,-2,0,2,-2,1,0,-1,1,0,-1,2,-3,3,-2,1,0,0,-1,1,0,-1
%N Third differences of {Pi*n^2}, fractional part of Pi*n^2.
%C From _Robert Israel_, Aug 20 2017: (Start)
%C -3 <= a(n) <= 3.
%C It appears that:
%C a(n) = -3 if and only if a(n+1) = 3.
%C if a(n) = -2 then a(n+1) = 0, 1 or 2.
%C if a(n) = -1 or 0 then a(n+1) = -1, 0, 1 or 2.
%C if a(n) = 1 then a(n+1) = -3, -1, 0 or 1.
%C if a(n) = 2 then a(n+1) = -3 or -2.
%C if a(n) = 3 then a(n+1) = -2 or -1.
%C (End)
%H Robert Israel, <a href="/A291196/b291196.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = {Pi*(n+3)^2} - 3*{Pi*(n+2)^2} + 3*{Pi*(n+1)^2} - {Pi*n^2}.
%p A0:= [seq(frac(Pi*n^2),n=1..103)]:
%p A1:= A0[2..-1]-A0[1..-2]:
%p A2:= A1[2..-1]-A1[1..-2]:
%p A2[2..-1]-A2[1..-2]; # _Robert Israel_, Aug 20 2017
%t Differences[FractionalPart[Pi*Range[100]^2], 3] (* _Paolo Xausa_, Feb 23 2024 *)
%K sign
%O 1,4
%A _Simon Plouffe_, Aug 20 2017
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