%I A000073 M1074 N0406
%S A000073 0,0,1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,
%T A000073 35890,66012,121415,223317,410744,755476,1389537,2555757,4700770,
%U A000073 8646064,15902591,29249425,53798080,98950096,181997601,334745777
%N A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=0, 
               a(2)=1.
%C A000073 Also (for n>2) number of ways writing 2^(n-2) as a product of decimal 
               digits of some other number which has no digits equal to 1; e.g. 
               n=8: 2^n=256, solutions = {488, ..., 8822, ..84222, .., 822222, ...4222222, 
               22222222}, their number is 81; so a(n+2)=A067374(2^n) - Labos E. 
               (labos(AT)ana.sote.hu), Jan 28 2002.
%C A000073 Also (for n>1) number of ordered trees with n+1 edges and having all 
               leaves at level three. Example: a(4)=2 because we have two ordered 
               trees with 5 edges and having all leaves at level three: (i) one 
               edge emanating from the root, at the end of which two paths of length 
               two are hanging and (ii) one path of length two emanating from the 
               root, at the end of which three edges are hanging. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Jan 03 2004
%C A000073 a(n)=number of compositions of n-2 with no part greater than 3. Example: 
               a(5)=4 because we have 1+1+1=1+2=2+1=3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Mar 10 2004
%C A000073 Let A=[0,0,1;1,1,1;0,1,0]. A000073(n) corresponds to both the (1,2) and 
               (3,1) positions in A^n. - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004
%C A000073 Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-2, with k=1, 
               r=2. - Vladimir Baltic (baltic(AT)matf.bg.ac.yu), Jan 17 2005
%C A000073 Number of binary sequences of length n-3 that have no three consecutive 
               0's. Example: a(7)=13 because among the 16 binary sequences of length 
               4 only 0000, 0001 and 1000 have 3 consecutive 0's. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Apr 27 2006
%C A000073 Let C = the tribonacci constant, 1.83928675...; then C^n = a(n)*(1/C) 
               + a(n+1)*(1/C + 1/C^2) + a(n+2)*(1/C + 1/C^2 + 1/C^3). Example: C^4 
               = 11.444...= 2*(1/C) + 4*(1/C + 1/C^2) + 7*(1/C + 1/C^2 + 1/C^3). 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 05 2006
%C A000073 a(n) =(j*c^n)+(k*r1^n)+(l*r2^n) where c is the Tribonacci constant (c=1,
               8392867552), real root of x^3-x^2-x-1=0 and r1 and r2 the two others 
               roots (complex) r1=m+pI r2=m-pI where m= (1-c)/2 (m=-0,4196433776) 
               and p = ((3*c-5)*(c+1)/4)^(1/2) (p=0,6062907292) and where j= 1/((c-m)^2+p^2) 
               (=0,1828035330) k = a+bI and l =a-bI where a= -j/2 (a=-0,0914017665) 
               and b=(c-m)/(2*p*((c-m)^2+p^2)(b=0,3405465308) - Philippe LALLOUET 
               (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
%C A000073 Convolved with the Padovan sequence = row sums of triangle A153462. [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]
%C A000073 For n>1: row sums of the triangle in A157897. [From Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Jun 25 2009]
%D A000073 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, p. 47, ex. 4.
%D A000073 M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, 
               Computers Math. Applic., 29 (No, 12, 1995), 103-110.
%D A000073 Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by 
               Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), 
               Article 06.1.8.
%D A000073 M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
%D A000073 M. Feinberg, New slants, Fib. Quart., 2 (1964), 223-227.
%D A000073 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
%D A000073 M. D. Hirschhorn, Coupled third-order recurrences, Fib. Quart., 44 (2006), 
               26-31.
%D A000073 O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular 
               automata, Comm. Math. Physics, 93 (1984), pp. 219-258, Reprinted 
               in Theory and Applications of Cellular Automata, S. Wolfram, Ed., 
               World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: 
               Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113. 
               See Eq. 5.5b.
%D A000073 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.
%D A000073 J. Riordan, An Introduction to Combinatorial Analysis, Princeton University 
               Press, Princeton, NJ, 1978.
%D A000073 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000073 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000073 M. E. Waddill and L. Sacks, Another generalized Fibonacci sequence, Fib. 
               Quart., 5 (1967), 209-222.
%H A000073 T. D. Noe, <a href="b000073.txt">Table of n, a(n) for n = 0..200</a>
%H A000073 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000073 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000073 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000073 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=10">
               Encyclopedia of Combinatorial Structures 10</a>
%H A000073 S. Kak, <a href="http://uk.arXiv.org/abs/physics/0411195">The Golden 
               Mean and the Physics of Aesthetics</a>
%H A000073 T. Mansour, <a href="http://arXiv.org/abs/math.CO/9909019">Permutations 
               avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in 
               S_4</a>
%H A000073 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000073 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000073 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Fibonaccin-StepNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000073 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TribonacciNumber.html">Link to a section of The World of Mathematics.</
               a>
%F A000073 G.f.: x^2/(1 - x - x^2 - x^3)
%F A000073 a(n+1)/a(n) -> A058265.
%F A000073 a(n) = center term in M^n * [1 0 0] where M = the 3X3 matrix [0 1 0 / 
               0 0 1 / 1 1 1]. (M^n * [1 0 0] = [a(n-1) a(n) a(n+1)]). 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
%F A000073 a(n+2)=sum{k=0..n, T(n-k, k)}, T(n, k) = trinomial coefficients (A027907); 
               - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005
%F A000073 A001590(n)=a(n+1)-a(n); A001590(n)=a(n-1)+a(n-2) for n>1; a(n)=(A000213(n+1)-A000213(n))/
               2; A000213(n-1)=a(n+2)-a(n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 22 2006
%F A000073 a(n)=3*c*((1/3)*(a+b+1))^n/(c^2-2*c+4) where a=(19+3*sqrt33)^(1/3), b=(19-3*sqrt33)^(1/
               3), c=(586+102*sqrt33)^(1/3). The offset is 1. a(3)=2. Round off 
               to the nearest integer.[From Al Hakanson (hawkuu(AT)gmail.com), Feb 
               02 2009]
%p A000073 A000073:=-z/(-1+z+z**2+z**3); [S. Plouffe in his 1992 dissertation.]
%t A000073 CoefficientList[Series[x^2/(1 - x - x^2 - x^3), {x, 0, 50}], x]
%o A000073 (PARI) {a(n) = polcoeff( if( n<0, x / ( 1 + x + x^2 - x^3), x^2 / ( 1 
               - x - x^2 - x^3) ) + x*O(x^abs(n)), abs(n))} /* Michael Somos Sep 
               03 2007 */
%o A000073 sage: from sage.combinat.sloane_functions import recur_gen3 sage: it 
               = recur_gen3(0,0,1,1,1,1) sage: [it.next() for i in range(38)] - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 24 2008
%Y A000073 Cf. A000213, A001590, A081172, A145027, A001644
%Y A000073 Cf. A063401, A008937, A089068, A027084.
%Y A000073 Cf. A062544, A077902, A054668, A027083, A027024.
%Y A000073 Cf. A118390.
%Y A000073 A057597 is this sequence run backwards: A057597(n) = a(1-n).
%Y A000073 Row 3 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A000073 A153462, A000931 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 
               2008]
%Y A000073 Sequence in context: A107281 A006744 A054175 this_sequence A160254 A005318 
               A102111
%Y A000073 Adjacent sequences: A000070 A000071 A000072 this_sequence A000074 A000075 
               A000076
%K A000073 nonn,easy,nice
%O A000073 0,5
%A A000073 N. J. A. Sloane (njas(AT)research.att.com).
%E A000073 More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000