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