|
| |
|
|
A001644
|
|
a(n)=a(n-1)+a(n-2)+a(n-3), a(0)=3, a(1)=1, a(2)=3.
(Formerly M2625 N1040)
|
|
61
| |
|
|
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, 995591267, 1831177831
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
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
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1,1).
|
|
|
FORMULA
| 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.
G.f.: g(x)=(3-2*x-x^2)/(1-x-x^2-x^3) - Miklos Kristof (kristmikl(AT)freemail.hu), Jul 29 2002
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]
|
|
|
MAPLE
| 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.]
|
|
|
MATHEMATICA
| f[x_] := f[x] = f[x - 1] + f[x - 2] + f[x - 3]; f[0] = 3; f[1] = 1; f[2] = 3
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]
LinearRecurrence[{1, 1, 1}, {3, 1, 3}, 60] (* From Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
|
|
|
PROG
| (PARI) a(n)=if(n<0, 0, polsym(1+x+x^2-x^3, n)[n+1])
|
|
|
CROSSREFS
| 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 * A139123 A133580 A019603
Adjacent sequences: A001641 A001642 A001643 * A001645 A001646 A001647
|
|
|
KEYWORD
| nonn,easy,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| 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
|
| |
|
|
