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
From R. J. Mathar, Sep 11 2008: (Start)
G.f.: (1+4*x) / (1-5*x+x^2).
(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
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Oct 28 2004
STATUS
approved