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

1,2

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.

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=1..24.

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

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.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A302762 A140796 A197212 * A270062 A270661 A222908

Adjacent sequences: A100056 A100057 A100058 * A100060 A100061 A100062

Keyword

nonn

Author

Gary W. Adamson, Oct 31 2004

Extensions

Edited by Ralf Stephan, Nov 02 2004

Status

approved