PROG
(MAGMAMagma) I:=[0, 4]; [n le 2 select I[n] else 5*Self(n-1) +4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2018
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Revision History for A106567
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
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reviewed
approved
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proposed
reviewed
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editing
proposed
MAPLE
with(linalg): M:=matrix(2, 2, [0, 4, 1, 5]): v[0]:=matrix(2, 1, [0, 1]): for n from 1 to 20 do v[n]:=multiply(M, v[n-1]) od: seq(v[n][1, 1], n=0..30);
MATHEMATICA
M={{0, 4}, {1, 5}}; v[1]= {0, 1}; v[n_]:= v[n]= M.v[n-1]; Table[v[n][[1]], {n, 30}]
KEYWORD
nonn,easy,less,changed
STATUS
proposed
editing
STATUS
editing
proposed
NAME
a(0)=0, a(1)=4; for n>1, a(n) = 5*a(n-1) + 4*a(n-2), with a(0) = 4, a(1) = 4.
COMMENTS
Old name was "First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,4],[1,5]] and v is the column vector [0,1]".
Real Pisot roots (the eigenvalues of M): -0.701562, 5.70156.
FORMULA
a(n) = 4*A015537(n).
a(n) = 4*(p^n - q^n)/(p - q), where 2*p = 5 + sqrt(41), 2*q = 5 - sqrt(41). - G. C. Greubel, Sep 06 2021
MAPLE
with(linalg): M:=matrix(2, 2, [0, 4, 1, 5]): v[0]:=matrix(2, 1, [0, 1]): for n from 1 to 20 do v[n]:=multiply(M, v[n-1]) od: seq(v[n][1, 1], n=0..2030);
MATHEMATICA
M = {{0, 4}, {1, 5}} ; v[1] = {0, 1} ; v[n_] := v[n] = M.v[n - 1] a = ; Table[v[n][[1]], {n, 1, 5030}]
CoefficientList[Series[4 *x / (1 - 5 *x - 4 *x^2), {x, 0, 2530}], x] (* Vincenzo Librandi, Mar 22 2018 *)
PROG
(MAGMA) I:=[0, 4, 20]; [n le 3 2 select I[n] else 5*Self(n-1) +4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2018
(Sage)
def A106567_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 4*x/(1-5*x-4*x^2) ).list()
A106567_list(30) # G. C. Greubel, Sep 06 2021
STATUS
approved
editing
STATUS
reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed