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%I #24 Sep 08 2022 08:45:01
%S 1,10,29,77,202,529,1385,3626,9493,24853,65066,170345,445969,1167562,
%T 3056717,8002589,20951050,54850561,143600633,375951338,984253381,
%U 2576808805,6746173034,17661710297,46238957857,121055163274
%N a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=10.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H G. C. Greubel, <a href="/A055850/b055850.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) = (10*(((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+7*x)/(1-3*x+x^2).
%F a(n) = Lucas(2n+1) + 6*Fibonacci(2n).
%F a(n) = Fibonacci(2*n+2) + 7*Fibonacci(2*n). - _G. C. Greubel_, Jan 16 2020
%p with(combinat); seq( fibonacci(2*n+2) + 7*fibonacci(2*n), n=0..30); # _G. C. Greubel_, Jan 16 2020
%t LinearRecurrence[{3,-1},{1,10},30] (* _Harvey P. Dale_, Jul 22 2019 *)
%o (Magma) [Lucas(2*n+1) + 6*Fibonacci(2*n): n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+7*x)/(1-3*x+x^2) )); // _Marius A. Burtea_, Jan 16 2020
%o (PARI) vector(31, n, fibonacci(2*n) + 7*fibonacci(2*(n-1)) ) \\ _G. C. Greubel_, Jan 16 2020
%o (Sage) [fibonacci(2*n+2) + 7*fibonacci(2*n) for n in (0..30)] # _G. C. Greubel_, Jan 16 2020
%o (GAP) List([0..30], n-> Fibonacci(2*n+2) + 7*Fibonacci(2*n) ); # _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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