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

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Keywords

• nonn
• easy

References

• 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
• Martin H. Gutknecht and Lloyd N. Trefethen, Real Polynomial Chebyshev Approximation by the Caratheodory-Fejer Method, SIAM Journal on Numerical Analysis, Vol. 19, No. 2 (Apr., 1982), pp. 358-371.
• Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

Crossrefs

Cf. A122099, A122100.

Programs

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

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

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 *)

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

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

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

Extensions

Edited by N. J. A. Sloane, Dec 04 2006

More terms from Colin Barker, Oct 18 2013