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%I #30 Jan 01 2024 11:53:27
%S 1,1,7,37,199,1069,5743,30853,165751,890461,4783807,25699957,
%T 138067399,741736909,3984819343,21407570533,115007491351,617852597821,
%U 3319277971807,17832095054677,95799031216999,514659346194349
%N a(n) = 5*a(n-1) + 2*a(n-2), n > 1; a(0) = a(1) = 1.
%C Top left element of powers of the matrix [1,2;3,4].
%H G. C. Greubel, <a href="/A124610/b124610.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,2).
%F a(n)/a(n-1) tends to (sqrt(33) + 5)/2 = 5.37228132... - _Gary W. Adamson_, Mar 03 2008
%F G.f.: (1 - 4*x)/(1 - 5*x - 2*x^2). - _G. C. Greubel_, Oct 23 2019
%e a(5) = 1069 because [1,2;3,4]^5 = [1069,1558; 2337,3406].
%p seq(coeff(series((1-4*x)/(1-5*x-2*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 23 2019
%t Table[MatrixPower[{{1, 2}, {3, 4}}, n][[1]][[1]], {n, 0, 30}]
%t Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{2,5},#]}]&, {1,1},40]][[1]] (* _Harvey P. Dale_, Mar 23 2011 *)
%t LinearRecurrence[{5,2},{1,1},30] (* _Harvey P. Dale_, Jan 01 2014 *)
%o (PARI) Vec((1-4*x)/(1-5*x-2*x^2) +O('x^30)) \\ _G. C. Greubel_, Oct 23 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x)/(1-5*x-2*x^2) )); // _G. C. Greubel_, Oct 23 2019
%o (Magma) [n le 2 select 1 else 5*Self(n-1) + 2*Self(n-2):n in [1..22]];// _Marius A. Burtea_, Oct 24 2019
%o (Sage)
%o def A124610_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-4*x)/(1-5*x-2*x^2) ).list()
%o A124610_list(30) # _G. C. Greubel_, Oct 23 2019
%o (GAP) a:=[1,1];; for n in [3..30] do a[n]:=5*a[n-1]+2*a[n-2]; od; a; # _G. C. Greubel_, Oct 23 2019
%Y Cf. A100638.
%K easy,nonn
%O 0,3
%A _Fredrik Johansson_, Dec 20 2006
%E Recurrence from _Gary W. Adamson_, Mar 03 2008
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