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A124610
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a(n) = 5*a(n-1) + 2*a(n-2), n > 1; a(0) = a(1) = 1.
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4
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1, 1, 7, 37, 199, 1069, 5743, 30853, 165751, 890461, 4783807, 25699957, 138067399, 741736909, 3984819343, 21407570533, 115007491351, 617852597821, 3319277971807, 17832095054677, 95799031216999, 514659346194349
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OFFSET
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0,3
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COMMENTS
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Top left element of powers of the matrix [1,2;3,4].
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LINKS
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FORMULA
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a(n)/a(n-1) tends to (sqrt(33) + 5)/2 = 5.37228132... - Gary W. Adamson, Mar 03 2008
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EXAMPLE
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a(5) = 1069 because [1,2;3,4]^5 = [1069,1558; 2337,3406].
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MAPLE
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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
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MATHEMATICA
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Table[MatrixPower[{{1, 2}, {3, 4}}, n][[1]][[1]], {n, 0, 30}]
Transpose[NestList[Flatten[{Rest[#], ListCorrelate[{2, 5}, #]}]&, {1, 1}, 40]][[1]] (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{5, 2}, {1, 1}, 30] (* Harvey P. Dale, Jan 01 2014 *)
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PROG
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(PARI) Vec((1-4*x)/(1-5*x-2*x^2) +O('x^30)) \\ G. C. Greubel, Oct 23 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x)/(1-5*x-2*x^2) )); // G. C. Greubel, Oct 23 2019
(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
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-4*x)/(1-5*x-2*x^2) ).list()
(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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