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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
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<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
(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
Sequence in context: A009174 A178345 A183547 * A357161 A290902 A155117
KEYWORD
nonn,easy
AUTHOR
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.)