The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123941 The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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
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
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 *)
PROG
(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
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+x^3))) \\ G. C. Greubel, Aug 05 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 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
CROSSREFS
Sequence in context: A000243 A076264 A018919 * A005774 A273343 A101169
KEYWORD
nonn,easy,less
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Nov 07 2006
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 18 16:58 EDT 2024. Contains 372664 sequences. (Running on oeis4.)