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%I #16 Mar 03 2017 08:31:20
%S 0,1,1,0,0,2,4,4,4,8,16,24,32,48,80,128,192,288,448,704,1088,1664,
%T 2560,3968,6144,9472,14592,22528,34816,53760,82944,128000,197632,
%U 305152,471040,727040,1122304,1732608,2674688,4128768,6373376,9838592,15187968,23445504
%N (1,3)-entry of the 3 X 3 matrix M^n, where M = {{0, -1, 1}, {1, 1, 0}, {0, 1, 1}}.
%C Essentially the same as A078003: a(n) = A078003(n-1).
%H Colin Barker, <a href="/A122788/b122788.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2).
%F Recurrence relation a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) (follows from the minimal polynomial of the matrix M).
%F a(n) = A078003(n-1). - _R. J. Mathar_, Aug 02 2008
%F G.f.: x*(1 - x) / (1 - 2*x + 2*x^2 - 2*x^3). - _Colin Barker_, Mar 03 2017
%e a(7)=4 because M^7 = {{0,4,4},{4,4,8},{8,12,12}}.
%p with(linalg): M[1]:=matrix(3,3,[0,-1,1,1,1,0,0,1,1]): for n from 2 to 42 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,3],n=1..42);
%p a[0]:=0: a[1]:=1: a[2]:=1: for n from 3 to 42 do a[n]:=2*a[n-1]-2*a[n-2]+2*a[n-3] od: seq(a[n],n=0..42);
%t M = {{0, -1, 1}, {1, 1, 0}, {0, 1, 1}}; v[1] = {0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
%o (PARI) concat(0, Vec(x*(1 - x) / (1 - 2*x + 2*x^2 - 2*x^3) + O(x^50))) \\ _Colin Barker_, Mar 03 2017
%K nonn,easy
%O 0,6
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 20 2006
%E Edited by _N. J. A. Sloane_, Nov 24 2006
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