A099867 a(n) = 5*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=9.

1, 9, 44, 211, 1011, 4844, 23209, 111201, 532796, 2552779, 12231099, 58602716, 280782481, 1345309689, 6445765964, 30883520131, 147971834691, 708975653324, 3396906431929, 16275556506321, 77980876099676, 373628823992059, 1790163243860619, 8577187395311036

Offset

0,2

Comments

From Klaus Purath, Mar 07 2023: (Start)

For any two terms (a(n), a(n+1)) = (x, y), x^2 - 5*x*y + y^2 = 37 = A082111(4). This is valid in general for all recursive sequences (t) with constant coefficients (5,-1) and t(0) = 1: x^2 - 5*x*y + y^2 = A082111(t(1)-5). This includes and interprets the Feb 04 2014 comment in A004253 by Colin Barker.

By analogy to all this, for three consecutive terms (x, y, z) of any sequence (t) of the form (5,-1) with t(0) = 1: y^2 - x*z = A082111(t(1)-5). (End)

Links

Colin Barker, Table of n, a(n) for n = 0..1000

A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (5,-1).

Formula

|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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A099868, A003501, A004254.

Sequence in context: A349878 A034558 A144109 * A228603 A297491 A104470

Adjacent sequences: A099864 A099865 A099866 * A099868 A099869 A099870

Keyword

nonn,easy

Author

Creighton Dement, Oct 28 2004

Status

approved