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%I #20 Sep 08 2022 08:45:16
%S 1,12,83,486,2645,13872,71303,362346,1829225,9198612,46150523,
%T 231225006,1157542205,5791962552,28972567343,144901100466,
%U 724620293585,3623445841692,18118262329763,90594411012726,452981353155365,2264934660052032,11324756983085783
%N a(n) = (19*5^n - 16*3^n + 1) / 4.
%C Sum of the entries in the last row of the 3 X 3 matrix M^n, where M = {{1, 0, 0}, {2, 3, 0}, {3, 4, 5}}.
%C Sum of the entries in the second row of M^n = A048473(n).
%H Colin Barker, <a href="/A102105/b102105.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 (9,-23,15).
%F a(n) = 9*a(n-1) - 23*a(n-2) + 15*a(n-3), a(0)=1,a(1)=12,a(2)=83 (derived from the minimal polynomial of the matrix M).
%F G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)). - _Colin Barker_, Mar 03 2017
%F E.g.f.: (exp(x) - 16*exp(3*x) + 19*exp(5*x))/4. - _G. C. Greubel_, Oct 27 2019
%e a(4) = 2645 = 9*486 - 23*83 + 15*12 = 9*a(3) - 23*a(2) + 15*a(1).
%e a(4) = 2645 since M^4 * {1, 1, 1} = {1, 161, 2645}, where 161 = A048473(4).
%p with(linalg): M[1]:=matrix(3,3,[1,0,0,2,3,0,3,4,5]): for n from 2 to 23 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(multiply(M[n],matrix(3,1,[1,1,1]))[3,1],n=1..23);
%p seq((19*5^n -16*3^n +1)/4, n=0..30); # _G. C. Greubel_, Oct 27 2019
%t Table[(19*5^n -16*3^n +1)/4, {n,0,30}] (* _G. C. Greubel_, Oct 27 2019 *)
%t LinearRecurrence[{9,-23,15},{1,12,83},30] (* _Harvey P. Dale_, Sep 19 2021 *)
%o (PARI) Vec((1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)) + O(x^30)) \\ _Colin Barker_, Mar 03 2017
%o (Magma) [(19*5^n -16*3^n +1)/4: n in [0..30]]; // _G. C. Greubel_, Oct 27 2019
%o (Sage) [(19*5^n -16*3^n +1)/4 for n in (0..30)] # _G. C. Greubel_, Oct 27 2019
%o (GAP) List([0..30], n-> (19*5^n -16*3^n +1)/4); # _G. C. Greubel_, Oct 27 2019
%Y Cf. A000326, A048473, A094727.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Dec 30 2004
%E Corrected by _T. D. Noe_, Nov 07 2006
%E Edited by _N. J. A. Sloane_, Dec 02 2006
%E New definition from _Ralf Stephan_, May 17 2007
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