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%I #11 Mar 02 2017 09:21:42
%S 1,6,19,73,264,973,3565,13086,48007,176149,646296,2371321,8700553,
%T 31923030,117128107,429752305,1576795176,5785386229,21227039605,
%U 77883687150,285761407807,1048481205661,3846960466104,14114802199681,51788325586033,190015462424934
%N The (1,1)-entry in the 3 X 3 matrix M^n, where M = [1,2,1 / 2,2,0 / 1,0,0].
%C Sequence generated from level 2 of the Pascal tetrahedron.
%D Peter Hilton, Derek Holton and Jean Pederson, "Mathematical Vistas, From a Room With Many Windows"; Springer, 2000, p. 178, Fig. 14, "The Pascal Tetrahedron".
%H Colin Barker, <a href="/A100191/b100191.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-2).
%F a(n) = 3*a(n-1) + 3*a(n-2) - 2*a(n-3) (derived from the minimal polynomial of the matrix M).
%F G.f.: x*(1 + 3*x - 2*x^2) / (1 - 3*x - 3*x^2 + 2*x^3). - _Colin Barker_, Mar 02 2017
%e a(4) = 73 because M^4 = [73,86,19 / 86,104,24 / 19,24,6]. Alternatively, a(4) = 3*a(3) + 3*a(2) - 2*a(1) = 57+18-2 = 73.
%p with(linalg): M[1]:=matrix(3,3,[1,2,1,2,2,0,1,0,0]): for n from 2 to 27 do M[n]:=multiply(M[1],M[n-1]) od: seq(M[n][1,1],n=1..27);
%p a[1]:=1: a[2]:=6: a[3]:=19: for n from 4 to 27 do a[n]:=3*a[n-1]+3*a[n-2]-2*a[n-3] od: seq(a[n],n=1..27);
%o (PARI) Vec(x*(1 + 3*x - 2*x^2) / (1 - 3*x - 3*x^2 + 2*x^3) + O(x^30)) \\ _Colin Barker_, Mar 02 2017
%Y Cf. A100190.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Nov 07 2004
%E Corrected by _T. D. Noe_, Nov 07 2006
%E Edited by _N. J. A. Sloane_, Dec 04 2006
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