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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

1. The other 3 columns are offsets of 1, 3, 7, 21, 59, ... starting with 1's. 2. 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/(2Cos k*Pi/9), k = 4.

LINKS

Table of n, a(n) for n=1..29.

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

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) [From Harvey P. Dale, Feb 17 2012]

EXAMPLE

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...

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 *)

PROG

(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

CROSSREFS

Cf. A066170, A046854.

Sequence in context: A104779 A178718 A244897 * A096240 A182887 A035080

Adjacent sequences:  A091647 A091648 A091649 * A091651 A091652 A091653

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, Jan 25 2004

EXTENSIONS

More terms from Harvey P. Dale, Feb 17 2012

STATUS

approved

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Last modified February 18 21:20 EST 2020. Contains 332028 sequences. (Running on oeis4.)