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).

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

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<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

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