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%I #50 Sep 08 2022 08:45:01
%S 1,9,26,69,181,474,1241,3249,8506,22269,58301,152634,399601,1046169,
%T 2738906,7170549,18772741,49147674,128670281,336863169,881919226,
%U 2308894509,6044764301,15825398394,41431430881,108468894249
%N a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=9.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H G. C. Greubel, <a href="/A055849/b055849.txt">Table of n, a(n) for n = 0..1000</a>
%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) = (9*(((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 G.f.: (1+6*x)/(1-3*x+x^2).
%F a(n) = L(2*n-1) + 2*L(2*n+1), where L(n) is the n-th Lucas number. - _Rigoberto Florez_, Dec 24 2018
%F a(n) = Fibonacci(2*n+2) + 6*Fibonacci(2*n). - _G. C. Greubel_, Jan 16 2020
%p with(combinat); seq( fibonacci(2*n+2) + 6*fibonacci(2*n), n=0..30); # _G. C. Greubel_, Jan 16 2020
%t LinearRecurrence[{3,-1},{1,9},30] (* _Harvey P. Dale_, Jan 20 2013 *)
%t Table[LucasL[2n-1]+2LucasL[2n+1], {n,0,30}] (* _Rigoberto Florez_, Dec 24 2018 *)
%o (PARI) vector(31, n, fibonacci(2*n) +6*fibonacci(2*n-2) ) \\ _G. C. Greubel_, Jan 16 2020
%o (Magma) [Lucas(2*n-1) + 2*Lucas(2*n+1): n in [0..30]]; // _G. C. Greubel_, Jan 16 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 26); Coefficients(R!( (1+6*x)/(1-3*x+x^2) )); // _Marius A. Burtea_, Jan 16 2020
%o (Sage) [fibonacci(2*n+2) + 6*fibonacci(2*n) for n in (0..30)] # _G. C. Greubel_, Jan 16 2020
%o (GAP) List([0..30], n-> Lucas(1,-1,2*n-1)[2] + 2*Lucas(1,-1,2*n+1)[2] ); # _G. C. Greubel_, Jan 16 2020
%Y Cf. A000032, A000045.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jun 03 2000
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