A122822 - OEIS

%I #12 Mar 03 2017 08:31:33

%S 0,0,-1,0,0,-1,1,2,3,10,19,35,71,131,240,446,810,1467,2660,4792,8621,

%T 15501,27814,49873,89384,160079,286589,512943,917813,1641978,2937132,

%U 5253248,9395035,16801268,30044388,53724067,96064297,171769178,307129259,549150614,981877515,1755576755,3138916347

%N The (1,4) entry in the matrix M^n, where M is the 4 X 4 matrix [[0,-1,1,0],[0,0,-1,1],[1,1,1,0],[0,1,1,1]].

%H Colin Barker, <a href="/A122822/b122822.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1,-3).

%F a(n) = 2*a(n-1) + a(n-3) - 3*a(n-4) for n>=4; a(0)=0, a(1)=-1, a(2)=0, a(3)=0.

%F G.f.: -x^2*(1 - 2*x) / (1 - 2*x - x^3 + 3*x^4). - _Colin Barker_, Mar 03 2017

%p a[0]:=0: a[1]:=0: a[2]:=-1: a[3]:=0: for n from 4 to 42 do a[n]:=2*a[n-1]+a[n-3]-3*a[n-4] od: seq(a[n],n=0..42);

%t M = {{0, -1, 1, 0}, {0, 0, -1, 1}, {1, 1, 1, 0}, {0, 1, 1, 1}}; v[1] = {0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

%o (PARI) concat(vector(2), Vec(-x^2*(1 - 2*x) / (1 - 2*x - x^3 + 3*x^4) + O(x^50))) \\ _Colin Barker_, Mar 03 2017

%K sign,easy

%O 0,8

%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 20 2006

%E Edited by _N. J. A. Sloane_, Oct 26 2006