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 A001644 a(n)=a(n-1)+a(n-2)+a(n-3), a(0)=3, a(1)=1, a(2)=3. (Formerly M2625 N1040) 65

%I M2625 N1040

%S 3,1,3,7,11,21,39,71,131,241,443,815,1499,2757,5071,9327,17155,31553,

%T 58035,106743,196331,361109,664183,1221623,2246915,4132721,7601259,

%U 13980895,25714875,47297029,86992799,160004703,294294531,541292033,995591267,1831177831

%N a(n)=a(n-1)+a(n-2)+a(n-3), a(0)=3, a(1)=1, a(2)=3.

%D M. Elia. "Derived Sequences, The Tribonacci Recurrence and Cubic Forms." The Fibonacci Quarterly 39.2 (2001): 107-109

%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

%D Fielder, Daniel C.; Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001644/b001644.txt">Table of n, a(n) for n=0..200</a>

%H G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3

%H G. Everest, Y. Puri and T. Ward, <a href="http://arXiv.org/abs/math.NT/0204173">Integer sequences counting periodic points</a>

%H A. Ilic, S. Klavzar, Y. Rho, <a href="http://dx.doi.org/10.2298/AADM120108002I">Generalized Lucas Cubes</a>, Appl. An. Disc. Math. 6 (2012) 82-94, proposition 11 for the sequence starting 1, 2, 4, 7, 11,...

%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>

%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (1,1,1).

%F Binet's formula: a(n)=r1^n+r2^n+r3^n, where r1, r2, r3 are the roots of the characteristic polynomial 1+x+x^2-x^3, see A058265.

%F a(n) = A000073(n)+2*A000073(n-1)+3*A000073(n-2).

%F G.f.: g(x)=(3-2*x-x^2)/(1-x-x^2-x^3) - _Miklos Kristof_, Jul 29 2002

%F a(n)=n*sum(k=1..n, sum(j=n-3*k..k, binomial(j,n-3*k+2*j)*binomial(k,j))/k), n>0, a(0)=3. [From _Vladimir Kruchinin_, Feb 24 2011]

%p A001644:=-(1+2*z+3*z**2)/(z**3+z**2+z-1); [_Simon Plouffe_ in his 1992 dissertation. Gives sequence except for the initial 3.]

%t f[x_] := f[x] = f[x - 1] + f[x - 2] + f[x - 3]; f[0] = 3; f[1] = 1; f[2] = 3

%t f[n_] := n*Sum[ Sum[ Binomial[j, n - 3*k + 2*j]*Binomial[k, j], {j, n - 3*k, k}]/k, {k, n}]; f[0] = 3; Array[f, 34, 0]

%t LinearRecurrence[{1, 1, 1}, {3, 1, 3}, 60] (* From _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)

%o (PARI) a(n)=if(n<0,0,polsym(1+x+x^2-x^3,n)[n+1])

%Y Cf. A000073, A106293 (Pisano periods)

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_.

%E Edited by Mario Catalani (mario.catalani(AT)unito.it), Jul 17 2002

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