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%I #70 Sep 08 2022 08:46:19
%S 2,2,9,16,37,72,149,296,597,1192,2389,4776,9557,19112,38229,76456,
%T 152917,305832,611669,1223336,2446677,4893352,9786709,19573416,
%U 39146837,78293672,156587349,313174696,626349397,1252698792,2505397589,5010795176,10021590357
%N a(0) = 2, a(1) = 2; for n > 1, a(n) = a(n-1) + 2*a(n-2) + 3.
%C Ratio of successive terms approaches 2.
%H Iain Fox, <a href="/A290604/b290604.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2)
%F a(n) = (2^(n+2) + 2*(-1)^n)/3 + 2^n - (3-(-1)^n)/2.
%F a(n) = A014113(n+1) + A141023(n).
%F G.f.: (2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3).
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n > 2. - _Iain Fox_, Dec 18 2017
%e a(0) = 2.
%e a(1) = 2.
%e a(2) = 2 + 2*2 + 3 = 9.
%e a(3) = 9 + 2*2 + 3 = 16.
%e a(4) = 16 + 9*2 + 3 = 37.
%e ...
%t Table[(2^(n + 2) + 2 (-1)^n) / 3 + 2^n - (3 - (-1)^n) / 2, {n, 0, 40}] (* _Vincenzo Librandi_, Oct 20 2017 *)
%o (PARI) Vec((2/(1-x-2*x^2)) + (3*x^2/((1-x)*(1-x-2*x^2))) + O(x^50)) \\ _Michel Marcus_, Oct 12 2017
%o (PARI) first(n) = Vec((2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3) + O(x^n)) \\ _Iain Fox_, Dec 18 2017
%o (Magma) [(2^(n+2)+2*(-1)^n)/3+2^n-(3-(-1)^n)/2: n in [0..40]]; // _Vincenzo Librandi_, Oct 20 2017
%K easy,nonn
%O 0,1
%A _Iain Fox_, Oct 11 2017
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