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A100058 Expansion of 1 / (1 - 3x - x^2 + 2x^3). 3

%I

%S 1,3,10,31,97,302,941,2931,9130,28439,88585,275934,859509,2677291,

%T 8339514,25976815,80915377,252043918,785093501,2445493667,7617486666,

%U 23727766663,73909799321,230222191294,717120839877,2233765112283

%N Expansion of 1 / (1 - 3x - x^2 + 2x^3).

%C a(n)/a(n-1) tends to 3.1149075414..., which is an eigenvalue of the matrix M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

%D Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications", Fibonacci Association, 1993, p. 27.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, 1, -2).

%F Recurrence: a(0) = 1, a(1) = 3, a(2) = 10; a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).

%F Given Hosoya's triangle: 1; 1, 1; 2, 1, 2; considered as an upper triangular 3 X 3 matrix M: [2 1 2 / 1 1 0 / 1 0 0]; a(n) = center term in M^n * [1 0 0].

%e a(5) = 97, center term in M^5 * [1 0 0]: [205 97 66].

%t CoefficientList[Series[1/(1 - 3x - x^2 + 2x^3), {x, 0, 25}], x] (* Or *)

%t Table[(MatrixPower[{{2, 1, 2}, {1, 1, 0}, {1, 0, 0}}, n].{1, 0, 0})[[2]], {n, 26}] (* _Robert G. Wilson v_, Nov 04 2004 *)

%t LinearRecurrence[{3,1,-2},{1,3,10},30] (* _Harvey P. Dale_, Mar 28 2012 *)

%o (PARI) Vec(1/(1-3*x-x^2+2*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%Y Partial sums of A052911. Cf. A019481, A052550, A052939, A100059, A058071.

%K nonn,easy

%O 0,2

%A _Gary W. Adamson_, Oct 31 2004

%E Edited by _Ralf Stephan_, Nov 02 2004

%E Corrected and extended by _Robert G. Wilson v_, Nov 04 2004

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)