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A073313 Binomial transform of generalized Lucas numbers S(n) = S(n-1) + S(n-2) + S(n-3), S(0)=3, S(1)=1, S(2)=3. 2
3, 4, 8, 22, 64, 184, 524, 1488, 4224, 11992, 34048, 96672, 274480, 779328, 2212736, 6282592, 17838080, 50647424, 143802560, 408296704, 1159271424, 3291504000, 9345523712, 26534621696, 75339399936, 213910160384, 607352285184 (list; graph; refs; listen; history; text; internal format)
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
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.
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
(Sage) ((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
Cf. A001644.
Sequence in context: A051440 A101932 A204521 * A217248 A168382 A155701
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Jul 26 2002
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)