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A001590
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Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=0, a(1)=1, a(2)=0.
(Formerly M0784 N0296)
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41
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0, 1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, 423, 778, 1431, 2632, 4841, 8904, 16377, 30122, 55403, 101902, 187427, 344732, 634061, 1166220, 2145013, 3945294, 7256527, 13346834, 24548655, 45152016, 83047505, 152748176, 280947697, 516743378, 950439251
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Dimensions of the homogeneous components of the higher order peak algebra associated to cubic roots of unity (Hilbert series = 1+1*t+2*t^2+3*t^3+6*t^4+11*t^5 ...) - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 22 2006
Starting with offset 3: (1, 2, 3, 6, 11, 10, 37,...) = row sums of triangle A145579. [From Gary W. Adamson, Oct 13 2008]
Starting (1, 2, 3, 6, 1l,...) = INVERT transform of the periodic sequence (1, 1, 0, 1, 1, 0, 1, 1, 0,...). [From Gary W. Adamson, May 04 2009]
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REFERENCES
| M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
M. Feinberg, New slants, Fib. Quart., 2 (1964), 223-227.
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).
M. E. Waddill and L. Sacks, Another generalized Fibonacci sequence, Fib. Quart., 5 (1967), 209-222.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 401
D. Krob and J.-Y. Thibon, Higher order peak algebras
Eric Weisstein's World of Mathematics, Tribonacci Number
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| G.f.: x*(1-x)/(1-x-x^2-x^3).
Limit a(n)/a(n-1)=x where x^3=1+x+x^2, x=1.839286755.... Let T(n)=A000073=0, 0, 1, 1, 2, 4, 7, 13... x^0=1 and for n>0 x^n=T(n-1)+a(n)*x+T(n)*x^2.
a(3*n) = Sum_{k+l+m=n} (n!/k!l!m!)*a(l+2*m). Example: a(12)=a(8)+4a(7)+10a(6)+16a(5)+19a(4)+16a(3)+10a(2)+4a(1)+a(0) The coefficients are the trinomial coefficients. T(n) and T(n-1) also satisfy this equation. (T(-1)=1)
a(n)=A000073(n+1)-A000073(n); a(n)=A000073(n-1)+A000073(n-2) for n>1; A000213(n-2)=a(n+1)-a(n) for n>1. - Reinhard Zumkeller, May 22 2006
a(n)+a(n+1)=A000213(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2006
If p[1]=0, p[i]=2, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1)=det A. [From Milan R. Janjic (agnus(AT)blic.net), May 02 2010]
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EXAMPLE
| a(12)=a(11)+a(10)+a(9): 230=125+68+37
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MATHEMATICA
| a=0; b=1; c=0; lst={a, b, c}; Do[d=a+b+c; AppendTo[lst, d]; a=b; b=c; c=d, {n, 5!}]; lst [From Vladimir Orlovsky, Sep 30 2008]
Join[{0}, LinearRecurrence[{1, 1, 1}, {1, 0, 1}, 40]] (* From Harvey P. Dale, Jan 19 2012 *)
LinearRecurrence[{1, 1, 1}, {0, 1, 0}, 50] (* From Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
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CROSSREFS
| Cf. A000045, A000073, A027907, A001590.
Cf. A027053, A078042.
A145579 [From Gary W. Adamson, Oct 13 2008]
Sequence in context: A010033 A065615 A054182 * A078042 A115792 A054177
Adjacent sequences: A001587 A001588 A001589 * A001591 A001592 A001593
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Henry Bottomley (se16(AT)btinternet.com), Jun 26 2001
Additional comments from Miklos Kristof (kristmikl(AT)freemail.hu), Jul 03 2002
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