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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
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
Eric Weisstein's World of Mathematics, Weyl Groups
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
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