A102105 a(n) = (19*5^n - 16*3^n + 1) / 4.

1, 12, 83, 486, 2645, 13872, 71303, 362346, 1829225, 9198612, 46150523, 231225006, 1157542205, 5791962552, 28972567343, 144901100466, 724620293585, 3623445841692, 18118262329763, 90594411012726, 452981353155365, 2264934660052032, 11324756983085783

Offset

0,2

Comments

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}}.

Sum of the entries in the second row of M^n = A048473(n).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Formula

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).

G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)). - Colin Barker, Mar 03 2017

E.g.f.: (exp(x) - 16*exp(3*x) + 19*exp(5*x))/4. - G. C. Greubel, Oct 27 2019

Example

a(4) = 2645 = 9*486 - 23*83 + 15*12 = 9*a(3) - 23*a(2) + 15*a(1).
a(4) = 2645 since M^4 * {1, 1, 1} = {1, 161, 2645}, where 161 = A048473(4).

Maple

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);
seq((19*5^n -16*3^n +1)/4, n=0..30); # G. C. Greubel, Oct 27 2019

Mathematica

Table[(19*5^n -16*3^n +1)/4, {n, 0, 30}] (* G. C. Greubel, Oct 27 2019 *)
LinearRecurrence[{9, -23, 15}, {1, 12, 83}, 30] (* Harvey P. Dale, Sep 19 2021 *)

Prog

(PARI) Vec((1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)) + O(x^30)) \\ Colin Barker, Mar 03 2017
(Magma) [(19*5^n -16*3^n +1)/4: n in [0..30]]; // G. C. Greubel, Oct 27 2019
(Sage) [(19*5^n -16*3^n +1)/4 for n in (0..30)] # G. C. Greubel, Oct 27 2019
(GAP) List([0..30], n-> (19*5^n -16*3^n +1)/4); # G. C. Greubel, Oct 27 2019

Crossrefs

Cf. A000326, A048473, A094727.

Sequence in context: A290715 A175037 A252179 * A275743 A026949 A165127

Adjacent sequences: A102102 A102103 A102104 * A102106 A102107 A102108

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 30 2004

Extensions

Corrected by T. D. Noe, Nov 07 2006

Edited by N. J. A. Sloane, Dec 02 2006

New definition from Ralf Stephan, May 17 2007

Status

approved