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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 4 05:49 EST 2023. Contains 367541 sequences. (Running on oeis4.)