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%I #16 Sep 08 2022 08:45:28
%S 0,1,3,15,71,340,1626,7778,37205,177966,851280,4072001,19477953,
%T 93170570,445670811,2131815570,10197297001,48777608903,233322137235,
%U 1116069871981,5338593130960,25536552265626,122151189577128,584296304368075,2794914830384226
%N The (1,4)-entry in the 4 X 4 matrix M^n, where M={{3, 2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}} (n>=0).
%D Martin H. Gutknecht and Lloyd N. Trefethen, Real Polynomial Chebyshev Approximation by the Caratheodory-Fejer Method, http://links.jstor.org/sici?sici=0036-1429(198204)19%3A2%3C358%3ARPCABT%3E2.0.CO%3
%D Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26
%H G. C. Greubel, <a href="/A123942/b123942.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-1,-1).
%F a(n) = 4*a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n>=4 (follows from the minimal polynomial of the matrix M).
%F G.f.: x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4). - _Colin Barker_, Oct 18 2013
%p with(linalg): M[1]:=matrix(4,4,[3,2,1,1,2,1,1,0,1,1,0,0,1,0,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0, seq(M[n][1,4], n=1..30);
%p a[0]:=0: a[1]:=1: a[2]:=3: a[3]:=15: for n from 4 to 30 do a[n]:=4*a[n-1] +4*a[n-2]-a[n-3]-a[n-4] od: seq(a[n], n=0..30);
%t M = {{3,2,1,1}, {2,1,1,0}, {1,1,0,0}, {1,0,0,0}}; v[1] = {0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
%t LinearRecurrence[{4,4,-1,-1}, {0,1,3,15}, 30] (* _G. C. Greubel_, Aug 05 2019 *)
%o (PARI) concat([0], Vec(x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4) + O(x^30))) \\ _Colin Barker_, Oct 18 2013
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4) )); // _G. C. Greubel_, Aug 05 2019
%o (Sage) (x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 05 2019
%o (GAP) a:=[0,1,3,15];; for n in [5..30] do a[n]:=4*a[n-1]+4*a[n-2]-a[n-3] -a[n-4]; od; a; # _G. C. Greubel_, Aug 05 2019
%Y Cf. A122099, A122100.
%K nonn,easy
%O 0,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2006
%E Edited by _N. J. A. Sloane_, Dec 04 2006
%E More terms from _Colin Barker_, Oct 18 2013
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