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%I #40 Apr 09 2023 02:24:19
%S 1,9,44,211,1011,4844,23209,111201,532796,2552779,12231099,58602716,
%T 280782481,1345309689,6445765964,30883520131,147971834691,
%U 708975653324,3396906431929,16275556506321,77980876099676,373628823992059,1790163243860619,8577187395311036
%N a(n) = 5*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=9.
%C From _Klaus Purath_, Mar 07 2023: (Start)
%C For any two terms (a(n), a(n+1)) = (x, y), x^2 - 5*x*y + y^2 = 37 = A082111(4). This is valid in general for all recursive sequences (t) with constant coefficients (5,-1) and t(0) = 1: x^2 - 5*x*y + y^2 = A082111(t(1)-5). This includes and interprets the Feb 04 2014 comment in A004253 by Colin Barker.
%C By analogy to all this, for three consecutive terms (x, y, z) of any sequence (t) of the form (5,-1) with t(0) = 1: y^2 - x*z = A082111(t(1)-5). (End)
%H Colin Barker, <a href="/A099867/b099867.txt">Table of n, a(n) for n = 0..1000</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/9-3/horadam-a.pdf">Pell Identities</a>, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).
%F |2*a(n) + A099868(n) - A003501(n+1)| = 20*A004254(n).
%F From _R. J. Mathar_, Sep 11 2008: (Start)
%F G.f.: (1+4*x) / (1-5*x+x^2).
%F a(n) = A004254(n+1) + 4*A004254(n).
%F (End)
%F a(n) = 2^(-1-n)*((5-sqrt(21))^n*(-13+sqrt(21)) + (5+sqrt(21))^n*(13+sqrt(21))) / sqrt(21). - _Colin Barker_, Mar 31 2017
%t a[0] = 1; a[1] = 9; a[n_] := a[n] = 5 a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 21}] (* _Robert G. Wilson v_, Dec 14 2004 *)
%t LinearRecurrence[{5, -1}, {1, 9}, 30] (* or *) CoefficientList[Series[(1 + 4 x)/(1 - 5 x + x^2), {x, 0, 30}], x] (* _Harvey P. Dale_, Jun 26 2011 *)
%o (Magma) I:=[1,9]; [n le 2 select I[n] else 5*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jul 30 2015
%o (PARI) Vec((1+4*x) / (1-5*x+x^2) + O(x^30)) \\ _Colin Barker_, Mar 31 2017
%Y Cf. A099868, A003501, A004254.
%K nonn,easy
%O 0,2
%A _Creighton Dement_, Oct 28 2004
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