OFFSET
0,1
COMMENTS
Limit as n-> infinity of a(n)/a(n-1) is 1+c, where c = 1.83928675...
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
H. Prodinger, Some information about the binomial transform., The Fibonacci Quarterly, 32, 1994, 412-415.
Index entries for linear recurrences with constant coefficients, signature (4,-4,2).
FORMULA
a(n) is the trace of the n-th power of 3 X 3 matrix: first row (2, 1, 0), second row (1, 1, 1), third row (1, 0, 1). It satisfies recurrence a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3), a(0)=3, a(1)=4, a(2)=8.
G.f.: (3 - 8*x + 4*x^2)/(1 - 4*x + 4*x^2 - 2*x^3).
MATHEMATICA
f[n_]:= f[n]=4*f[n-1]-4*f[n-2]+2*f[n-3]; f[0]=3; f[1]=4; f[2]=8; Table[f[n], {n, 0, 30}]
LinearRecurrence[{4, -4, 2}, {3, 4, 8}, 30] (* Harvey P. Dale, May 08 2015 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((3-8*x+4*x^2)/(1-4*x+4*x^2-2*x^3)) \\ G. C. Greubel, Apr 22 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (3-8*x+4*x^2)/(1-4*x+4*x^2-2*x^3) )); // G. C. Greubel, Apr 22 2019
(SageMath) ((3-8*x+4*x^2)/(1-4*x+4*x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Jul 26 2002
STATUS
approved