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

1, 3, 10, 31, 97, 302, 941, 2931, 9130, 28439, 88585, 275934, 859509, 2677291, 8339514, 25976815, 80915377, 252043918, 785093501, 2445493667, 7617486666, 23727766663, 73909799321, 230222191294, 717120839877, 2233765112283

Offset

0,2

Comments

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.

References

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

Links

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (3, 1, -2).

Formula

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

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].

Example

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

Mathematica

CoefficientList[Series[1/(1 - 3x - x^2 + 2x^3), {x, 0, 25}], x] (* Or *)
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 *)
LinearRecurrence[{3, 1, -2}, {1, 3, 10}, 30] (* Harvey P. Dale, Mar 28 2012 *)

Prog

(PARI) Vec(1/(1-3*x-x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Crossrefs

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

Sequence in context: A055217 A097472 A068094 * A002160 A214839 A114487

Adjacent sequences: A100055 A100056 A100057 * A100059 A100060 A100061

Keyword

nonn,easy

Author

Gary W. Adamson, Oct 31 2004

Extensions

Edited by Ralf Stephan, Nov 02 2004

Corrected and extended by Robert G. Wilson v, Nov 04 2004

Status

approved