A052911 -id:A052911 - cp's OEIS frontend

A100059 First differences of A052911.

1, 5, 14, 45, 139, 434, 1351, 4209, 13110, 40837, 127203, 396226, 1234207, 3844441, 11975078, 37301261, 116189979, 361921042, 1127350583, 3511592833, 10938286998, 34071752661, 106130359315, 330586256610

Offset: 1

Gary W. Adamson, Oct 31 2004

nonn

a(n)/a(n-1) tends to 3.11490754148...an eigenvalue of M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

			a(5) = 139 = rightmost term in M^5 * [1 1 1] which is [434 205 139]. 434 = a(6), while 205 = A052911(5).
a(6) = 434 = 3*a(5) + a(4) - 2*a(3) = 3*139 + 45 - 2*14.

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

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.

Cf. A019481, A052550, A052939, A100058, A058071.

LinearRecurrence[{3,1,-2},{1,5,14},30] (* Harvey P. Dale, Apr 21 2016 *)

G.f.: (2*x^2-2*x-1)*x / (-2*x^3+x^2+3*x-1).
Recurrence: a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = rightmost term in M^5 * [1 1 1], where M = the 3 X 3 upper triangular matrix [2 1 2 / 1 1 0 / 1 0 0].
INVERT transform of (1, 4, 5, 6, 7, 8, 9, ...) with offset 0.

Edited by Ralf Stephan, Nov 02 2004

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

Original entry on oeis.org

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

Gary W. Adamson, Oct 31 2004

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.

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

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

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.

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

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 *)

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

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].

Edited by Ralf Stephan, Nov 02 2004

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

cp's OEIS Frontend

A100059 First differences of A052911.

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