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a(n) = ((5+sqrt(3))*(2+sqrt(3))^n + (5-sqrt(3))*(2-sqrt(3))^n)/2.
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%I #16 Sep 08 2022 08:45:46

%S 5,13,47,175,653,2437,9095,33943,126677,472765,1764383,6584767,

%T 24574685,91713973,342281207,1277410855,4767362213,17792037997,

%U 66400789775,247811121103,924843694637,3451563657445,12881410935143,48074080083127

%N a(n) = ((5+sqrt(3))*(2+sqrt(3))^n + (5-sqrt(3))*(2-sqrt(3))^n)/2.

%C Binomial transform of A162562. Second binomial transform of A162813. Inverse binomial transform of A162814.

%H Harvey P. Dale, <a href="/A162563/b162563.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).

%F a(n) = 4*a(n-1) - a(n-2) for n > 1; a(0) = 5, a(1) = 13.

%F G.f.: (5-7*x)/(1-4*x+x^2).

%F a(n) = 4*a(n-1) - a(n-2), with a(0)=5 and a(1)=13. - _Paolo P. Lava_, Jul 15 2009

%t LinearRecurrence[{4,-1},{5,13},30] (* _Harvey P. Dale_, Aug 25 2014 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((5+r)*(2+r)^n+(5-r)*(2-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 14 2009

%Y Cf. A162562, A162813, A162814.

%K nonn

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 18 2009