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A102105
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a(n) = (19*5^n - 16*3^n + 1) / 4.
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1
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1, 12, 83, 486, 2645, 13872, 71303, 362346, 1829225, 9198612, 46150523, 231225006, 1157542205, 5791962552, 28972567343, 144901100466, 724620293585, 3623445841692, 18118262329763, 90594411012726, 452981353155365, 2264934660052032, 11324756983085783
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).
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FORMULA
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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
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EXAMPLE
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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).
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MAPLE
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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);
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PROG
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(PARI) Vec((1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)) + O(x^30)) \\ Colin Barker, Mar 03 2017
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CROSSREFS
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Cf. A000326, A094727, A048473.
Sequence in context: A290715 A175037 A252179 * A275743 A026949 A165127
Adjacent sequences: A102102 A102103 A102104 * A102106 A102107 A102108
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KEYWORD
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nonn,easy
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AUTHOR
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Gary W. Adamson, Dec 30 2004
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EXTENSIONS
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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
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STATUS
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approved
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