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Search: id:A000213
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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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+0 72
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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
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Number of (n-1)-bit binary sequences with each one adjacent to a zero. - Ron Hardin (rhhardin(AT)att.net), Dec 24 2007
The binomial transform is A099216. The inverse binomial transform is (-1)^n*A124395(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 19 2008]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)
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REFERENCES
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B. G. Baumgart, Letter to the editor, Fib. Quart. 2 (1964), 260, 302.
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
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
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T. D. Noe, Table of n, a(n) for n = 0..200
Joerg Arndt, Fxtbook
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
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FORMULA
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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 (qntmpkt(AT)yahoo.com), 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 (reinhard.zumkeller(AT)gmail.com), 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]
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MAPLE
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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
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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 (4vladimir(AT)gmail.com), Sep 30 2008]
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PROGRAM
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sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(1, 1, 1, 1, 1, 1) sage: [it.next() for i in range(35)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
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Cf. A000288, A000322, A000383, A046735, A060455.
Adjacent sequences: A000210 A000211 A000212 this_sequence A000214 A000215 A000216
Sequence in context: A102475 A066173 A114322 this_sequence A074858 A074860 A135728
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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