

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



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
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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", SpringerVerlag, 1989, p. 170, E.43: "Newton's Method According to Newton".


LINKS



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(n1) tends to 2.0945514...; e.g. a(12)/a(11) = 7249/3457 = 2.0969048...
Empirical: a(n) = 2*a(n2)+5*a(n3). G.f.: x*(1+7*x+5*x^2)/(12*x^25*x^3).  Colin Barker, Jan 26 2012
Empirical formula follows from the CayleyHamilton 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(n2)+5*a(n3), a(1)=1, a(2)=7, a(3)=7}, a(n), remember):


MATHEMATICA



CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



