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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.

%I A073313 #25 Jan 05 2025 19:51:37
%S A073313 3,4,8,22,64,184,524,1488,4224,11992,34048,96672,274480,779328,
%T A073313 2212736,6282592,17838080,50647424,143802560,408296704,1159271424,
%U A073313 3291504000,9345523712,26534621696,75339399936,213910160384,607352285184
%N 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.
%C A073313 Limit as n-> infinity of a(n)/a(n-1) is 1+c, where c = 1.83928675...
%D A073313 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.
%H A073313 G. C. Greubel, <a href="/A073313/b073313.txt">Table of n, a(n) for n = 0..1000</a>
%H A073313 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A073313 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A073313 H. Prodinger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-5/prodinger.pdf">Some information about the binomial transform.</a>, The Fibonacci Quarterly, 32, 1994, 412-415.
%H A073313 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,2).
%F A073313 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.
%F A073313 G.f.: (3 - 8*x + 4*x^2)/(1 - 4*x + 4*x^2 - 2*x^3).
%t A073313 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}]
%t A073313 LinearRecurrence[{4,-4,2},{3,4,8},30] (* _Harvey P. Dale_, May 08 2015 *)
%o A073313 (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
%o A073313 (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
%o A073313 (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
%Y A073313 Cf. A001644.
%K A073313 easy,nonn
%O A073313 0,1
%A A073313 Mario Catalani (mario.catalani(AT)unito.it), Jul 26 2002