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A247565 a(n) = 5*a(n-1) - 10*a(n-2) + 8*a(n-3) with a(0) = 2, a(1) = a(2) = 3.

%I A247565 #27 Dec 22 2023 14:13:07
%S A247565 2,3,3,1,-1,9,63,217,527,969,1311,1081,47,-87,7743,39961,121679,
%T A247565 270729,456543,548857,344687,-112791,380031,5785561,24225167,66310473,
%U A247565 135585183,208622521,217744559,87179049,-72570177,507315097,3959709647,14144835849,35185603551
%N A247565 a(n) = 5*a(n-1) - 10*a(n-2) + 8*a(n-3) with a(0) = 2, a(1) = a(2) = 3.
%H A247565 G. C. Greubel, <a href="/A247565/b247565.txt">Table of n, a(n) for n = 0..2500</a>
%H A247565 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,8).
%F A247565 G.f.: (2 - 7*x + 8*x^2) / (1 - 5*x + 10*x^2 - 8*x^3).
%F A247565 (n) = a(-1-n) * 2^(2*n+1) for all n in Z.
%F A247565 a(n) = 2^n + A247560(n) for all n in Z.
%F A247565 a(n) = A247564(n+1) * A247564(n) for all n in Z.
%F A247565 0 = a(n)*(+4*a(n+1) + 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z.
%e A247565 G.f. = 2 + 3*x + 3*x^2 + x^3 - x^4 + 9*x^5 + 63*x^6 + 217*x^7 + 527*x^8 + ...
%t A247565 CoefficientList[Series[(2-7*x+8*x^2)/(1-5*x+10*x^2-8*x^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{5,-10,8}, {2,3,3}, 60] (* _G. C. Greubel_, Aug 04 2018 *)
%o A247565 (PARI) {a(n) = 2^n + real( (1 + quadgen(-7))^n )};
%o A247565 (PARI) Vec((2 - 7*x + 8*x^2) / (1 - 5*x + 10*x^2 - 8*x^3) + O(x^50)) \\ _Michel Marcus_, Sep 22 2014
%o A247565 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-7*x+8*x^2)/(1-5*x+10*x^2-8*x^3))); // _G. C. Greubel_, Aug 04 2018
%Y A247565 Cf. A247560, A247564.
%K A247565 sign,easy
%O A247565 0,1
%A A247565 _Michael Somos_, Sep 20 2014