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.
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
Offset: 0
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).
Crossrefs
Cf. A001644.
Programs
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Magma
R
:=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 -
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 *) -
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 -
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
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).
Comments