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

Author

Gary W. Adamson, Oct 31 2004

Keywords

• nonn
• easy

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.

Examples

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

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

• Index entries for linear recurrences with constant coefficients, signature (3, 1, -2).
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.

Crossrefs

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

Programs

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 *)

PARI

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

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].

Extensions

Edited by Ralf Stephan, Nov 02 2004

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