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A099867
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a(n) = 5*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=9.
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2
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1, 9, 44, 211, 1011, 4844, 23209, 111201, 532796, 2552779, 12231099, 58602716, 280782481, 1345309689, 6445765964, 30883520131, 147971834691, 708975653324, 3396906431929, 16275556506321, 77980876099676, 373628823992059, 1790163243860619, 8577187395311036
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OFFSET
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0,2
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pps. 245-252.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (5,-1).
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FORMULA
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|2*a(n) + A099868(n) - A003501(n+1)| = 20*A004254(n).
From R. J. Mathar, Sep 11 2008: (Start)
G.f.: (1+4*x) / (1-5*x+x^2).
a(n) = A004254(n+1) + 4*A004254(n).
(End)
a(n) = 2^(-1-n)*((5-sqrt(21))^n*(-13+sqrt(21)) + (5+sqrt(21))^n*(13+sqrt(21))) / sqrt(21). - Colin Barker, Mar 31 2017
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MATHEMATICA
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a[0] = 1; a[1] = 9; a[n_] := a[n] = 5 a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 21}] (* Robert G. Wilson v, Dec 14 2004 *)
LinearRecurrence[{5, -1}, {1, 9}, 30] (* or *) CoefficientList[Series[(1 + 4 x)/(1 - 5 x + x^2), {x, 0, 30}], x] (* Harvey P. Dale, Jun 26 2011 *)
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PROG
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(MAGMA) I:=[1, 9]; [n le 2 select I[n] else 5*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 30 2015
(PARI) Vec((1+4*x) / (1-5*x+x^2) + O(x^30)) \\ Colin Barker, Mar 31 2017
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CROSSREFS
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Cf. A099868, A003501, A004254.
Sequence in context: A084903 A034558 A144109 * A228603 A104470 A282722
Adjacent sequences: A099864 A099865 A099866 * A099868 A099869 A099870
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KEYWORD
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nonn,easy
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AUTHOR
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Creighton Dement, Oct 28 2004
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STATUS
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approved
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