1,2

A sequence generated from a polynomial explored by Newton.

Barbeau quotes Isaac Newton's "Analysis by Equations of an Infinite Number of Terms", providing Newton's "Method" of finding the real root of x^3 - 2x - 5, in which Newton states "Finally, subducting the negative Part of the Quotient from the affirmative, I have 2.0945514... the Quotient sought".

E. J. Barbeau, "Polynomials", Springer-Verlag, 1989, p. 170, E.43: "Newton's Method According to Newton".

Robert Israel, Table of n, a(n) for n = 1..3111

Index entries for linear recurrences with constant coefficients, signature (0,2,5).

Given x^3 - 2x - 5, the real root (and convergent of the sequence), 2.0945514815... is an eigenvalue of the 3 X 3 matrix M.

a(n)/a(n-1) tends to 2.0945514...; e.g. a(12)/a(11) = 7249/3457 = 2.0969048...

Empirical: a(n) = 2*a(n-2)+5*a(n-3). G.f.: x*(1+7*x+5*x^2)/(1-2*x^2-5*x^3). - Colin Barker, Jan 26 2012

Empirical formula follows from the Cayley-Hamilton theorem. - Robert Israel, Sep 19 2019

a(5) = 49, the center term in M^n * [1 1 1] which = [19 49 73].

f:= gfun:-rectoproc({a(n)=2*a(n-2)+5*a(n-3), a(1)=1, a(2)=7, a(3)=7}, a(n), remember):

map(f, [$1..100]); # Robert Israel, Sep 19 2019

LinearRecurrence[{0, 2, 5}, {1, 7, 7}, 40] (* Jean-François Alcover, Aug 29 2022 *)

Sequence in context: A391659 A299453 A300091 * A282828 A339722 A341833

Adjacent sequences: A094245 A094246 A094247 * A094249 A094250 A094251

nonn,easy

Gary W. Adamson, Apr 24 2004

Corrected by T. D. Noe, Nov 07 2006

approved