A094248 Consider 3 X 3 matrix M = [0 1 0 / 0 0 1 / 5 2 0]; a(n) = the center term in M^n * [1 1 1].

1, 7, 7, 19, 49, 73, 193, 391, 751, 1747, 3457, 7249, 15649, 31783, 67543, 141811, 294001, 621337, 1297057, 2712679, 5700799, 11910643, 24964993, 52325281, 109483201, 229475527, 480592807, 1006367059, 2108563249, 4415698153, 9248961793, 19374212551, 40576414351

Offset

1,2

Comments

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".

References

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

Links

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

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

Formula

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

Example

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

Maple

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

Mathematica

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

Crossrefs

Sequence in context: A213031 A299453 A300091 * A282828 A339722 A341833

Adjacent sequences: A094245 A094246 A094247 * A094249 A094250 A094251

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 24 2004

Extensions

Corrected by T. D. Noe, Nov 07 2006

Status

approved