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A120665 a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) for n>3, a(1)=0, a(2)=-1, a(3)=0, 1
0, -1, 0, 9, 52, 231, 936, 3641, 13884, 52407, 196768, 736713, 2754180, 10288199, 38415000, 143404569, 535268812, 1997801751, 7456200336, 27827523881, 103854943764, 387594348327, 1446526643848, 5398520615673, 20147572596060, 75191803322999, 280619707804800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Old name was: "Sequence produced by a 3 X 3 Markov chain based on a Cartan matrix."

Characteristic polynomial is 2 - 9 x + 6 x^2 - x^3.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

MAGMA, MAGMA help file

Eric Weisstein's World of Mathematics, Weyl Groups

Index entries for linear recurrences with constant coefficients, signature (6,-9,2).

FORMULA

Recurrence (via the Cayley-Hamilton theorem): a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) (see the 2nd Maple program). - Emeric Deutsch, Jul 18 2007

O.g.f.: -x^2*(-1+6*x) / ((2*x-1)*(x^2-4*x+1)). - R. J. Mathar, Dec 05 2007

a(n) = (-2^(2+n) + (11-6*sqrt(3))*(2+sqrt(3))^n + (2-sqrt(3))^n*(11+6*sqrt(3))) / 6. - Colin Barker, Feb 05 2017

MAPLE

with(linalg): M := matrix(3, 3, [2, -1, 0, -1, 2, -2, 0, -1, 2]): v[1] := matrix(3, 1, [0, 1, 2]): for n from 2 to 25 do v[n] := multiply(M, v[n-1]) end do: seq(v[n][1, 1], n = 1 .. 25); # Emeric Deutsch, Jul 18 2007

a[1]:=0: a[2]:=-1: a[3]:=0: for n from 4 to 25 do a[n]:= 6*a[n-1]-9*a[n-2]+2*a[n-3] end do: seq(a[n], n=1..25); # Emeric Deutsch, Jul 18 2007

MATHEMATICA

M = {{2, -1, 0}, {-1, 2, -2}, {0, -1, 2}} ; v[1] = {0, 1, 2} ; v[n_] := v[n] = M.v[n - 1] ; a = Table[ v[n][[1]], {n, 1, 50}]

PROG

(PARI) concat(0, Vec(-x^2*(1 - 6*x) / ((1 - 2*x)*(1 - 4*x + x^2)) + O(x^30))) \\ Colin Barker, Feb 05 2017

CROSSREFS

Sequence in context: A152580 A197722 A172470 * A163941 A289418 A292488

Adjacent sequences:  A120662 A120663 A120664 * A120666 A120667 A120668

KEYWORD

sign,easy

AUTHOR

Roger L. Bagula, Aug 11 2006, corrected Jul 13 2007

EXTENSIONS

Edited by N. J. A. Sloane, Jul 13 2007, Jul 21 2007

New name from Joerg Arndt, Feb 05 2017

STATUS

approved

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Last modified June 1 01:40 EDT 2020. Contains 334758 sequences. (Running on oeis4.)