A123941 The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}.

0, 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639, 954538564968, 2748484256480

Offset: 0

Author

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

Keywords

• nonn
• easy
• less

Comments

Essentially the same as A076264. - Tom Edgar, May 12 2015

References

• Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..2000
• Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Crossrefs

Cf. A018919, A122099, A122100.

Programs

GAP

a:=[0,1,3];; for n in [4..30] do a[n]:=3*a[n-1]-a[n-3]; od; a; # Muniru A Asiru, Oct 28 2018

Magma

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/(1-3*x+x^3) )); // G. C. Greubel, Aug 05 2019

Maple

with(linalg): M[1]:=matrix(3,3,[2,1,1,1,1,0,1,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,2], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=3: for n from 3 to 30 do a[n]:=3*a[n-1]-a[n-3] od: seq(a[n], n=0..30);

Mathematica

M = {{2,1,1}, {1,1,0}, {1,0,0}}; v[1] = {0,0,1}; v[n_]:= v[n] =M.v[n-1];Table[v[n][[2]], {n, 30}]
LinearRecurrence[{3,0,-1}, {0,1,3}, 30] (* G. C. Greubel, Aug 05 2019 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+x^3))) \\ G. C. Greubel, Aug 05 2019

Sage

(x/(1-3*x+x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

Formula

a(n) = 3*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=3 (follows from the minimal polynomial x^3-3x^2+1 of the matrix M).

a(n) = A076264(n-1). - R. J. Mathar, Jun 18 2008

G.f.: x/(1 - 3*x + x^3). - Arkadiusz Wesolowski, Oct 29 2012

a(n) = A018919(n-2) for n >= 2. - Georg Fischer, Oct 28 2018

Extensions

Edited by N. J. A. Sloane, Nov 07 2006