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 A123942 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). 2
 0, 1, 3, 15, 71, 340, 1626, 7778, 37205, 177966, 851280, 4072001, 19477953, 93170570, 445670811, 2131815570, 10197297001, 48777608903, 233322137235, 1116069871981, 5338593130960, 25536552265626, 122151189577128, 584296304368075, 2794914830384226 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES 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 Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1). FORMULA 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). G.f.: x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4). - Colin Barker, Oct 18 2013 MAPLE 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); 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); MATHEMATICA 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}] LinearRecurrence[{4, 4, -1, -1}, {0, 1, 3, 15}, 30] (* G. C. Greubel, Aug 05 2019 *) PROG (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 (Magma) R:=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 (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 (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 CROSSREFS Cf. A122099, A122100. Sequence in context: A009174 A178345 A183547 * A357161 A290902 A155117 Adjacent sequences: A123939 A123940 A123941 * A123943 A123944 A123945 KEYWORD nonn,easy AUTHOR Roger L. Bagula and Gary W. Adamson, Oct 25 2006 EXTENSIONS Edited by N. J. A. Sloane, Dec 04 2006 More terms from Colin Barker, Oct 18 2013 STATUS approved

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Last modified May 29 03:48 EDT 2024. Contains 372921 sequences. (Running on oeis4.)