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A106707 First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-1],[1,4]] and v is the column vector [0,1]. 5

%I

%S 0,-1,-4,-15,-56,-209,-780,-2911,-10864,-40545,-151316,-564719,

%T -2107560,-7865521,-29354524,-109552575,-408855776,-1525870529,

%U -5694626340,-21252634831,-79315912984,-296011017105,-1104728155436,-4122901604639,-15386878263120,-57424611447841

%N First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-1],[1,4]] and v is the column vector [0,1].

%C a(n) is the first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, -1], [1, 4]] and v[0] is the column vector [0,1].

%C Real Pisot roots (the eigenvalues of M): 2-sqrt(3)=0.267949, 2+sqrt(3)=3.73205.

%H G. C. Greubel, <a href="/A106707/b106707.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).

%F G.f.: -x/(1-4*x+x^2).

%F a(n) = 4*a(n-1) - a(n-2); a(0)=0, a(1)=-1.

%F a(n) = (1/6)*sqrt(3)*[2-sqrt(3)]^n - (1/6)*sqrt(3)*[2+sqrt(3)]^n, with n>=0. - _Paolo P. Lava_, Oct 06 2008

%p a[0]:=0: a[1]:=-1: for n from 2 to 27 do a[n]:=4*a[n-1]-a[n-2] od: seq(a[n],n=0..27);

%t M = {{0, -1}, {1, 4}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}]

%t Table[Round[(1/6)*Sqrt[3]*(2 - Sqrt[3])^n - (1/6)*Sqrt[3]*(2 + Sqrt[3])^n ], {n,0,50}] (* _G. C. Greubel_, Feb 05 2018 *)

%t LinearRecurrence[{4,-1},{0,-1},30] (* _Harvey P. Dale_, Nov 01 2019 *)

%o (PARI) x='x+O('x^30); Vec(-x/(1-4*x+x^2)) \\ _G. C. Greubel_, Feb 05 2018

%o (MAGMA) I:=[0,-1]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Feb 05 2018

%Y Cf. A001076, A001353.

%K sign,easy

%O 0,3

%A _Roger L. Bagula_, May 30 2005

%E Edited by _N. J. A. Sloane_, Apr 30 2006

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Last modified May 6 09:45 EDT 2021. Contains 343580 sequences. (Running on oeis4.)