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A078112 Coefficients a(n) in the unique expansion sin(1) = Sum[a(n)/n!, n>=1], where a(n) satisfies 0<=a(n)

%I A078112 #11 Feb 16 2016 11:54:04
%S A078112 0,1,2,0,0,5,6,0,0,9,10,0,0,13,14,0,0,17,18,0,0,21,22,0,0,25,26,0,0,
%T A078112 29,30,0,0,33,34,0,0,37,38,0,0,41,42,0,0,45,46,0,0,49,50,0,0,53,54,0,
%U A078112 0,57,58,0,0,61,62,0,0,65,66,0,0,69,70,0,0,73,74,0,0,77,78,0,0,81,82,0,0,85
%N A078112 Coefficients a(n) in the unique expansion sin(1) = Sum[a(n)/n!, n>=1], where a(n) satisfies 0<=a(n)<n.
%H A078112 Colin Barker, <a href="/A078112/b078112.txt">Table of n, a(n) for n = 1..1000</a>
%H A078112 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,-1).
%F A078112 a(n) = floor(n!*sin(1)) - n*floor((n-1)!*sin(1)). a(n)=0 if n==0 or 1 (mod 4); a(n)=n-1 if n==2 or 3 (mod 4). - _Benoit Cloitre_, Dec 07 2002
%F A078112 From _Colin Barker_, Feb 15 2016: (Start)
%F A078112 a(n) = 2*a(n-1)-3*a(n-2)+4*a(n-3)-3*a(n-4)+2*a(n-5)-a(n-6) for n>6.
%F A078112 G.f.: x^2*(1-x^2+2*x^3) / ((1-x)^2*(1+x^2)^2). (End)
%e A078112 sum(i=1,10,a(i)/i!)=0.84147073..., sin(1)=0.841470984...
%o A078112 (PARI) concat(0, Vec(x^2*(1-x^2+2*x^3)/((1-x)^2*(1+x^2)^2) + O(x^100))) \\ _Colin Barker_, Feb 15 2016
%Y A078112 Cf. A077814.
%K A078112 nonn,easy
%O A078112 1,3
%A A078112 _John W. Layman_, Dec 04 2002
%E A078112 More terms from _Benoit Cloitre_, Dec 07 2002