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%I #17 Sep 08 2022 08:45:39
%S 1,7,54,448,3896,34832,316224,2894528,26609536,245174272,2261620224,
%T 20875015168,192738922496,1779844247552,16437306875904,
%U 151809149370368,1402086588645376,12949609668739072,119602725461950464,1104655331042787328,10202654714273202176
%N a(n) = ((7+sqrt(5))^n + (7-sqrt(5))^n)/2.
%H Vincenzo Librandi, <a href="/A152108/b152108.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 (14, -44).
%F From _Philippe Deléham_, Nov 26 2008: (Start)
%F a(n) = 14*a(n-1) - 44*a(n-2), n > 1; a(0)=1, a(1)=7.
%F G.f.: (1-7*x)/(1-14*x+44*x^2).
%F a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2*k)*5^(n-k))/7^n. (End)
%t With[{srt5=Sqrt[5]},Simplify/@Table[((7+srt5)^n+(7-srt5)^n)/2,{n,0,20}]] (* or *) LinearRecurrence[{14,-44},{1,7},20] (* _Harvey P. Dale_, Jan 16 2012 *)
%t CoefficientList[Series[(1 - 7 x) / (1 - 14 x + 44 x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 04 2018 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r5>:=NumberField(x^2-5); S:=[ ((7+r5)^n+(7-r5)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 26 2008
%o (Magma) I:=[1,7]; [n le 2 select I[n] else 14*Self(n-1)-44*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 04 2018
%K nonn
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008
%E Extended beyond a(6) by _Klaus Brockhaus_, Nov 26 2008
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