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%I #41 Sep 08 2022 08:45:01
%S 1,6,17,45,118,309,809,2118,5545,14517,38006,99501,260497,681990,
%T 1785473,4674429,12237814,32039013,83879225,219598662,574916761,
%U 1505151621,3940538102,10316462685,27008849953,70710087174,185121411569,484654147533,1268841031030
%N a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H G. C. Greubel, <a href="/A054492/b054492.txt">Table of n, a(n) for n = 0..1000</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%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 (3,-1).
%F a(n) = (6*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - (((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)))/sqrt(5).
%F a(n) = 2*Lucas(2*n+1) - Fibonacci(2*n+1).
%F G.f.: (1+3*x)/(1-3*x+x^2). - _Philippe Deléham_, Nov 03 2008
%F a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n-1). - _Ehren Metcalfe_, Mar 26 2016
%F E.g.f.: (1/10) * exp((3-sqrt(5))*x/2) * ((5-9*sqrt(5)) + (5+9*sqrt(5)) * exp(sqrt(5)*x) ). - _G. C. Greubel_, Mar 26 2016
%t CoefficientList[Series[(1 + 3 x) / (1 - 3 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 20 2015 *)
%t LinearRecurrence[{3, -1}, {1, 6}, 100] (* _G. C. Greubel_, Mar 26 2016 *)
%o (PARI) Vec((1+3*x)/(1-3*x+x^2) + O(x^30)) \\ _Michel Marcus_, Mar 20 2015
%o (Magma) I:=[1,6]; [n le 2 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 20 2015
%Y Cf. A002878, A054486.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 06 2000
%E More terms from _Vincenzo Librandi_, Mar 20 2015
%E Typo in name fixed by _Karl V. Keller, Jr._, Jun 23 2015
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