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Search: id:A001644
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| A001644 |
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a(n)=a(n-1)+a(n-2)+a(n-3), a(0)=3, a(1)=1, a(2)=3.
(Formerly M2625 N1040)
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+0
60
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| 3, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499, 2757, 5071, 9327, 17155, 31553, 58035, 106743, 196331, 361109, 664183, 1221623, 2246915, 4132721, 7601259, 13980895, 25714875, 47297029, 86992799, 160004703, 294294531, 541292033
(list; graph; listen)
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OFFSET
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0,1
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
M. Elia. "Derived Sequences, The Tribonacci Recurrence and Cubic Forms." The Fibonacci Quarterly 39.2 (2001): 107-109
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Fielder, Daniel C.; Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Index entries for sequences related to linear recurrences with constant coefficients
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Lucas n-Step Number
Eric Weisstein's World of Mathematics, Tribonacci Number
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FORMULA
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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.
G.f.: g(x)=(3-2*x-x^2)/(1-x-x^2-x^3) - Miklos Kristof (kristmikl(AT)freemail.hu), Jul 29 2002
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MAPLE
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A001644:=-(1+2*z+3*z**2)/(z**3+z**2+z-1); [S. Plouffe in his 1992 dissertation. Gives sequence except for the initial 3.]
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MATHEMATICA
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f[x_] := f[x] = f[x - 1] + f[x - 2] + f[x - 3]; f[0] = 3; f[1] = 1; f[2] = 3
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polsym(1+x+x^2-x^3, n)[n+1])
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CROSSREFS
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a(n) is related to the tribonacci numbers T(n) (A000073) by a(n)=T(n)+2*T(n-1)+3T(n-2).
Cf. A000073.
Sequence in context: A064434 A086401 A095732 this_sequence A139123 A133580 A019603
Adjacent sequences: A001641 A001642 A001643 this_sequence A001645 A001646 A001647
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Edited by Mario Catalani (mario.catalani(AT)unito.it), Jul 17 2002
Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
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