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

Author

Gary W. Adamson, Oct 31 2004

Keywords

• nonn

Comments

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.

Examples

			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.
		

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

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

Programs

Mathematica

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

Formula

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.

Extensions

Edited by Ralf Stephan, Nov 02 2004