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A133585
Expansion of x - x^2*(2*x+1)*(x^2-2) / ( (x^2-x-1)*(x^2+x-1) ).
3
1, 2, 4, 5, 10, 13, 26, 34, 68, 89, 178, 233, 466, 610, 1220, 1597, 3194, 4181, 8362, 10946, 21892, 28657, 57314, 75025, 150050, 196418, 392836, 514229, 1028458, 1346269, 2692538, 3524578, 7049156, 9227465, 18454930, 24157817
OFFSET
1,2
COMMENTS
A133585 is a companion to A133586.
FORMULA
Equals the matrix-matrix-vector product A133566 * A133080 * A000045 (previous name).
For even-indexed terms, a(n) = F(n+1). For odd-indexed terms (n>1), a(n) = 2*a(n-1), A126358.
EXAMPLE
a(4) = F(5) = 5.
a(5) = 2*a(4) = 2*5 = 10.
MAPLE
A133585aux := proc(n, k)
add(A133566(n, j)*A133080(j, k), j=k..n) ;
end proc:
A000045 := proc(n)
combinat[fibonacci](n) ;
end proc:
A133585 := proc(n)
add(A133585aux(n, j)*A000045(j), j=0..n) ;
end proc: # R. J. Mathar, Jun 20 2015
MATHEMATICA
CoefficientList[Series[1 - x (2 x + 1) (x^2 - 2)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{0, 3, 0, -1}, {1, 2, 4, 5, 10}, 40] (* Harvey P. Dale, Mar 04 2019 *)
PROG
(PARI) a(n)=if(n>1, ([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 3, 0]^(n-2)*[2; 4; 5; 10])[1, 1], 1) \\ Charles R Greathouse IV, Jun 20 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Sep 18 2007
EXTENSIONS
Previous name corrected and new name from R. J. Mathar, Jun 20 2015
STATUS
approved