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%I #26 Sep 08 2022 08:45:27
%S 0,9,17,72,209,711,2250,7357,23693,76848,248413,804307,2602122,
%T 8421705,27251521,88190472,285386041,923535567,2988612714,9671371877,
%U 31297187845,101279874144,327748481957,1060616489147,3432226859754
%N Expansion of x^2*(9 + 8*x - 8*x^2)/((1+x-x^2)*(1-2*x-4*x^2)).
%H G. C. Greubel, <a href="/A121955/b121955.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,2,-4).
%F a(n) = a(n-1) + 7*a(n-2) + 2*a(n-3) - 4*a(n-4). - _R. J. Mathar_, Apr 04 2009
%F a(n) = ((-1)^n*Fibonacci(n+2) + 2^n*(5*Fibonacci(n+1) - 4*Fibonacci(n)))/3. - _G. C. Greubel_, Oct 07 2019
%p with(combinat); f:=fibonacci; seq(((-1)^n*f(n+2) + 2^n*(5*f(n+1) - 4*f(n)))/3, n=1..30); # _G. C. Greubel_, Oct 07 2019
%t M = {{0,1,1,1,0,1}, {1,0,1,1,1,0}, {1,1,0,0,1,1}, {1,1,0,0,0,0}, {0,1,1, 0,0,0}, {1,0,1,0,0,0}}; v[1]={0,1,1,2,3,5}; v[n_]:= v[n]= M.v[n-1]; Table[Floor[v[n][[1]]], {n, 50}]
%t LinearRecurrence[{1,7,2,-4},{0,9,17,72},30] (* _Harvey P. Dale_, Jul 29 2013 *)
%t With[{F=fibonacci}, Table[((-1)^n*F[n+2] + 2^n*(5*F[n+1]-4*F[n]))/3, {n,30}]] (* _G. C. Greubel_, Oct 07 2019 *)
%o (PARI) vector(30, n, f=fibonacci; ((-1)^n*f(n+2) + 2^n*(5*f(n+1) - 4*f(n)))/3) \\ _G. C. Greubel_, Oct 07 2019
%o (Magma) F:=Fibonacci; [((-1)^n*F(n+2) + 2^n*(5*F(n+1) - 4*F(n)))/3: n in [1..30]]; // _G. C. Greubel_, Oct 07 2019
%o (Sage) f=fibonacci; [((-1)^n*f(n+2) + 2^n*(5*f(n+1) - 4*f(n)))/3 for n in (1..30)] # _G. C. Greubel_, Oct 07 2019
%o (GAP) F:=Fibonacci;; List([1..30], n-> ((-1)^n*F(n+2) + 2^n*(5*F(n+1) - 4*F(n)))/3); # _G. C. Greubel_, Oct 07 2019
%Y Cf. A000045.
%K nonn,easy
%O 1,2
%A _Roger L. Bagula_, Sep 01 2006
%E New name (g.f. given by Mathar) by _Joerg Arndt_, Feb 06 2013
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