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%I #18 Sep 08 2022 08:45:01
%S 1,11,40,117,319,848,2233,5859,15352,40205,105271,275616,721585,
%T 1889147,4945864,12948453,33899503,88750064,232350697,608302035,
%U 1592555416,4169364221,10915537255,28577247552,74816205409
%N a(n) = 3*a(n-1) - a(n-2) + 8 with a(0)=1, a(1)=11.
%H G. C. Greubel, <a href="/A056124/b056124.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F a(n) = ( 19*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - 9*(((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)) )/sqrt(5) - 8.
%F G.f.: (1+7*x)/((1-x)*(1-3*x+x^2)).
%F a(n) = Fibonacci(2*n+5) + 2*Lucas(2*n) - 8.
%F From _G. C. Greubel_, Jan 19 2020: (Start)
%F a(n) = Fibonacci(2*n+2) + 8*Fibonacci(2*n+1) - 8.
%F E.g.f.: exp(3*x/2)*( 9*cosh(sqrt(5)*x/2) - (11/sqrt(5))*sinh(sqrt(5)*x/2) ) - 8*exp(x). (End)
%p with(combinat); seq( fibonacci(2*n+2) + 8*fibonacci(2*n+1) - 8, n=0..30); # _G. C. Greubel_, Jan 19 2020
%t LinearRecurrence[{4,-4,1},{1,11,40},30] (* _Harvey P. Dale_, Mar 25 2015 *)
%o (PARI) vector(31, n, fibonacci(2*n) +8*fibonacci(2*n-1) -8 ) \\ _G. C. Greubel_, Jan 19 2020
%o (Magma) [Fibonacci(2*n+2) + 8*Fibonacci(2*n+1) - 8: n in [0..30]]; // _G. C. Greubel_, Jan 19 2020
%o (Sage) [fibonacci(2*n+2) + 8*fibonacci(2*n+1) - 8 for n in (0..30)] # _G. C. Greubel_, Jan 19 2020
%o (GAP) List([0..30], n-> Fibonacci(2*n+2) + 8*Fibonacci(2*n+1) - 8 ); # _G. C. Greubel_, Jan 19 2020
%Y Cf. A000032, A000045, A055850 (first differences).
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jul 07 2000
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