A091650 Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 -1 3 2]. Set seed vector = [1 1 1 1] = first row, then take M*[1 1 1 1] = [1 1 1 3] then M * [1 1 1 3], etc. Sequence gives terms in rightmost column.

1, 3, 7, 21, 59, 171, 491, 1415, 4073, 11729, 33771, 97241, 279993, 806209, 2321385, 6684163, 19246279, 55417453, 159568195, 459458307, 1322957467, 3809304207, 10968454313, 31582405473, 90937912211, 261845282321, 753953441489, 2170922412257, 6250921954449

Offset: 1

Author

Gary W. Adamson, Jan 25 2004

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to a 9-Gon diagonal.

The other 3 columns are offsets of 1, 3, 7, 21, 59, ... starting with 1's.

The characteristic equation of the 4 X 4 matrix is x^4 - 2x^3 - 3x^4 + x + 1 (coefficients may be found in A066170) with roots 2.879385241..., -1, -.5320888862... and .65270364466... An alternative matrix giving the same eigenvalues (refer to A046854) relates to the 9-Gon: [1 1 1 1 / 1 1 1 0 / 1 1 0 0 / 1 0 0 0] since the eigenvalue 2.8793852...is the longest diagonal of the 9-Gon given edge = 1. Or, 2.879385... = 1/(2*cos(k*Pi/9)), k = 4.

Examples

			a(5) = 59 since M*[1 1 1 1] then 4 iterates = [3 7 21 59]. a(5) = rightmost term.
a(10)/a(9) = 11729/4073 = 2.8796955...
		

Links

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

Crossrefs

Cf. A066170, A046854.

Programs

Mathematica

Rest[CoefficientList[Series[x (1+x-2x^2-x^3)/(1-2x-3x^2+x^3+x^4),{x,0,40}],x]] (* or *) LinearRecurrence[{2,3,-1,-1},{1,3,7,21},40] (* Harvey P. Dale, Feb 17 2012 *)

PARI

Vec((1+x-2*x^2-x^3)/(1-2*x-3*x^2+x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 31 2012

Formula

G.f.: x*(1+x-2*x^2-x^3)/(1-2*x-3*x^2+x^3+x^4). - Colin Barker, Jan 31 2012

a(1)=1, a(2)=3, a(3)=7, a(4)=21, a(n)=2*a(n-1)+3*a(n-2)-a(n-3)-a(n-4). - Harvey P. Dale, Feb 17 2012

Extensions

More terms from Harvey P. Dale, Feb 17 2012