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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.)