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A056123 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=11.

%I A056123 #20 Sep 08 2022 08:45:01
%S A056123 1,11,32,85,223,584,1529,4003,10480,27437,71831,188056,492337,1288955,
%T A056123 3374528,8834629,23129359,60553448,158530985,415039507,1086587536,
%U A056123 2844723101,7447581767,19498022200,51046484833,133641432299
%N A056123 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=11.
%H A056123 G. C. Greubel, <a href="/A056123/b056123.txt">Table of n, a(n) for n = 0..1000</a>
%H A056123 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A056123 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F A056123 a(n) = {11*[((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 A056123 G.f.: (1+8*x)/(1-3*x+x^2).
%F A056123 a(n) = 6*Lucas(2n+1) - Fibonacci(2n+5).
%F A056123 From _G. C. Greubel_, Jan 17 2020: (Start)
%F A056123 a(n) = Fibonacci(2*n+2) + 8*Fibonacci(2*n).
%F A056123 E.g.f.: exp(3*t/2)*( cosh(sqrt(5)*t/2) + (19/sqrt(5))*sinh(sqrt(5)*t/2) ). (End)
%p A056123 with(combinat); seq( fiboacci(2*n+2) +8*fibonacci(2*n), n=0..30); # _G. C. Greubel_, Jan 17 2020
%t A056123 Table[Fibonacci[2*n+2] +8*Fibonacci[2*n], {n,0,30}] (* _G. C. Greubel_, Jan 17 2020 *)
%o A056123 (PARI) vector(31, n, fibonacci(2*n) +8*fibonacci(2*n-2) ) \\ _G. C. Greubel_, Jan 17 2020
%o A056123 (Magma) [Fibonacci(2*n+2) +8*Fibonacci(2*n): n in [0..30]]; // _G. C. Greubel_, Jan 17 2020
%o A056123 (Sage) [fibonacci(2*n+2) +8*fibonacci(2*n) for n in (0..30)] # _G. C. Greubel_, Jan 17 2020
%o A056123 (GAP) List([0..30], n-> Fibonacci(2*n+2) +8*Fibonacci(2*n) ); # _G. C. Greubel_, Jan 17 2020
%Y A056123 Cf. A000032, A000045, A055850.
%K A056123 easy,nonn
%O A056123 0,2
%A A056123 _Barry E. Williams_, Jul 06 2000