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A000213
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Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=a(2)=1.
(Formerly M2454 N0975)
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86
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1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, 2209, 4063, 7473, 13745, 25281, 46499, 85525, 157305, 289329, 532159, 978793, 1800281, 3311233, 6090307, 11201821, 20603361, 37895489, 69700671, 128199521, 235795681, 433695873, 797691075, 1467182629
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of (n-1)-bit binary sequences with each one adjacent to a zero. - R. H. Hardin, Dec 24 2007
The binomial transform is A099216. The inverse binomial transform is (-1)^n*A124395(n). [From R. J. Mathar, Aug 19 2008]
Contribution from Gary W. Adamson, Apr 27 2009: (Start)
Equals INVERT transform of (1, 0, 2, 0, 2, 0, 2,...). a(6) = 17 =
(1, 1, 1, 3, 5, 9) dot (0, 2, 0, 2, 0, 1) = (0 + 2 + 0 + 6 + 0 + 9) = 17. (End)
Equals the number of tilings of a 2 X n grid using singletons and "S-shaped quadrominos" (i.e. shapes of the form Polygon[{{0, 0}, {2, 0}, {2, 1}, {3, 1}, {3, 2}, {1, 2}, {1, 1}, {0, 1}}]) - John M. Campbell, May 16, 2011.
Equals the number of tilings of a 2 X n grid using singletons and "T-shaped quadrominos" (i.e. shapes of the form Polygon[{{0,0},{3,0},{3,1},{2,1},{2,2},{1,2},{1,1},{0,1}}]) - John M. Campbell, May 16, 2011.
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REFERENCES
| B. G. Baumgart, Letter to the editor, Fib. Quart. 2 (1964), 260, 302.
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
Joanna Jaszunska and Jan Okninski, Structure of Chinese algebras, Journal of Algebra, Volume 346, Issue 1, 15 November 2011, Pages 31-81; http://www.sciencedirect.com/science/article/pii/S0021869311004698.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
Joerg Arndt, Fxtbook, p.312
Nick Hobson, Python program for this sequence
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, Tribonacci Number
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1,1).
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FORMULA
| G.f.: (1-x)*(1+x)/(1-x-x^2-x^3). - Ralf Stephan, Feb 11 2004
a(n) = rightmost term of M^n * [1 1 1], where M = the 3X3 matrix [1 1 1 / 1 0 0 / 0 1 0]. (M^n * [1 1 1]= [a(n+2) a(n+1) a(n)]). a(n)/a(n-1) tends to the tribonacci constant, 1.839286755...; an eigenvalue of M and a root of x^3 - x^2 - x - 1 = 0. - Gary W. Adamson, Dec 17 2004
a(n)=A001590(n+3)-A001590(n+2); a(n+1)-a(n)=2*A000073(n); a(n)=A000073(n+3)-A000073(n+1). - Reinhard Zumkeller, May 22 2006
a(n)=A001590(n)+A001590(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2006
a(n) ~ (F - 1) * T^n, where F = A086254 and T = A058265. [From Charles R Greathouse IV, Nov 09 2008]
a(n)=2*a(n-1)-a(n-4),n>3 [From Gary Detlefs, Sep 13 2010]
a(n)=sum(m=0..n/2, sum(i=0..m, binomial(n-2*m+1,m-i)*binomial(n-2*m+i,n-2*m))). [From Vladimir Kruchinin, Dec 17 2011]
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MAPLE
| K:=(1-z^2)/(1-z-z^2-z^3): Kser:=series(K, z=0, 45): seq((coeff(Kser, z, n)), n= 0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 08 2007
A000213:=(z-1)*(1+z)/(-1+z+z**2+z**3); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a=1; b=1; c=1; 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]
LinearRecurrence[{1, 1, 1}, {1, 1, 1}, 30] (* From Harvey P. Dale, May 23 2011 *)
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PROG
| (PARI) a(n)=n=[1, 1, 1; 1, 0, 0; 0, 1, 0]^n; n[3, 1]+n[3, 2]+n[3, 3] \\ Charles R Greathouse IV, Feb 18, 2011
(Maxima) a(n):=sum(sum(binomial(n-2*m+1, m-i)*binomial(n-2*m+i, n-2*m), i, 0, m), m, 0, (n)/2); [From Vladimir Kruchinin, Dec 17 2011]
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CROSSREFS
| Cf. A000288, A000322, A000383, A046735, A060455.
Sequence in context: A102475 A066173 A114322 * A074858 A074860 A171856
Adjacent sequences: A000210 A000211 A000212 * A000214 A000215 A000216
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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