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%I #33 Jul 27 2016 21:46:01
%S 1,-1,4,8,19,32,64,125,256,512,1027,2048,4096,8189,16384,32768,65539,
%T 131072,262144,524285,1048576,2097152,4194307,8388608,16777216,
%U 33554429,67108864,134217728,268435459,536870912,1073741824,2147483645,4294967296,8589934592
%N a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3, a(0)=1, a(1)=-1, a(2)=4, a(3)=8.
%C a(n) mod 9 = 1, 8, 4, 8, 1, 5, 1, 8, 4, 8, 1, 5, ... (repeat).
%C Difference table for a(n):
%C 1, -1, 4, 8, 19, 32, ...
%C -2, 5, 4, 11, 13, 32, ...
%C 7, -1, 7, 2, 19, 29, ...
%C -8, 8, -5, 17, 10, 41, ...
%C 16, -13, 22, -7, 31, 14, ...
%C -29, 35, -29, 38, -17, 65, ...
%C ... .
%C The recurrence of the name is valid for every line and the main diagonal which is in A014551.
%H Colin Barker, <a href="/A274817/b274817.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).
%F G.f.: (x^3+6*x^2-3*x+1) / (-2*x^4+x^3-2*x+1). - _Colin Barker_, Jul 07 2016
%F a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3. - _Wesley Ivan Hurt_, Jul 07 2016
%F a(n) = 2^n - sin(n*Pi/3)*(sqrt(3) + 2*sin(2*n*Pi/3)). - _Wesley Ivan Hurt_, Jul 07 2016
%F a(n) = 2^n - period 6: repeat [0, 3, 0, 0, -3, 0].
%F a(n+1) = 2*a(n) + period 6: repeat [-3, 6, 0, 3, -6, 0].
%F a(n+3) = -a(n) + 9*2^n.
%F a(n) = A014551(n) - A057079(n).
%p A274817:=n->2^n - sin(n*Pi/3)*(sqrt(3) + 2*sin(2*n*Pi/3)): seq(A274817(n), n=0..40); # _Wesley Ivan Hurt_, Jul 07 2016
%t Table[2^n - Sin[n*Pi/3] (Sqrt[3] + 2 Sin[2*n*Pi/3]), {n, 0, 40}] (* _Wesley Ivan Hurt_, Jul 07 2016 *)
%t LinearRecurrence[{2, 0, -1, 2}, {1, -1, 4, 8}, 100] (* _G. C. Greubel_, Jul 07 2016 *)
%o (PARI) Vec((x^3+6*x^2-3*x+1)/(-2*x^4+x^3-2*x+1) + O(x^40)) \\ _Colin Barker_, Jul 07 2016
%Y Cf. A000079, A014551, A057079.
%K sign,easy
%O 0,3
%A _Paul Curtz_, Jul 07 2016
%E One term corrected and more terms added by _Colin Barker_, Jul 07 2016
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