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A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=0, a(2)=1.
(Formerly M1074 N0406)
239
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080, 98950096, 181997601, 334745777, 615693474, 1132436852 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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, Jan 03 2004

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, Mar 10 2004

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, Oct 15 2004

Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-2, with k=1, r=2. - Vladimir Baltic, Jan 17 2005

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, Apr 27 2006

Therefore, the complementary sequence to A050231 (n coin tosses with a run of three heads). a(n) = 2^(n-3) - A050231(n-3) - Toby Gottfried, Nov 21 2010

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, Nov 05 2006

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 are the two other roots (complex), r1 = m+pI and 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

Convolved with the Padovan sequence = row sums of triangle A153462. - Gary W. Adamson, Dec 27 2008

For n>1: row sums of the triangle in A157897. - Reinhard Zumkeller, Jun 25 2009

a(n+2) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 0, 0, 1; 1, 0, 0] or [1, 1, 0; 1, 0, 1; 1, 0, 0] or [1, 1, 1; 1, 0, 0; 0, 1, 0] or [1, 0, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014

a(n-1) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 0, 1; 1, 1, 1; 0, 1, 0], [0, 1, 0; 0, 1, 1; 1, 1, 0], [0, 0, 1; 1, 0, 1; 0, 1, 1] or [0, 1, 0; 0, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

Also row sums of A082601 and of A082870. - Reinhard Zumkeller, Apr 13 2014

REFERENCES

Abrate, Marco; Barbero, Stefano; Cerruti, Umberto; Murru, Nadir. Colored compositions, Invert operator and elegant compositions with the "black tie''. Discrete Math. 335 (2014), 1--7. MR3248794

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 47, ex. 4.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.

Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

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.

J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.

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 and Simone Sandri, Table of n, a(n) for n = 0..3000 (first 200 terms by T. D. Noe)

Joerg Arndt, Matters Computational (The Fxtbook), pp.307-309

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135

D. Birmajer, J. B. Gil, M. D. Weiner, Linear recurrence sequences via Bell polynomials, arXiv preprint arXiv:1405.7727, 2014

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Dairyko, Michael; Tyner, Samantha; Pudwell, Lara; Wynn, Casey. Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From N. J. A. Sloane, Feb 01 2013

M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers Math. Applic., 29 (No, 12, 1995), 103-110.

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.

M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.

M. Feinberg, New slants, Fib. Quart. 2 (1964), 223-227.

M. D. Hirschhorn, Coupled third-order recurrences, Fib. Quart., 44 (2006), 26-31.

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 10

M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. - N. J. A. Sloane, Sep 16 2012

S. Kak, The Golden Mean and the Physics of Aesthetics

Vladimir Kruchinin Composition of ordinary generating functions

Steven Linton, James Propp, Tom Roby, Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.

T. Mansour, Permutations avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in S_4

T. Mansour and M. Shattuck, Polynomials whose coefficients are generalized Tribonacci numbers, Applied Mathematics and Computation, Volume 219, Issue 15, 1 April 2013, Pages 8366-8374.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013

J. L. Ramirez and V. F. Sirvent, Incomplete Tribonacci Numbers and Polynomials, Journal of Integer Sequences, Vol. 17, 2014, #14.4.2.

M. Shattuck, Combinatorial identities for incomplete tribonacci polynomials, arXiv preprint arXiv:1406.2755, 2014.

M. E. Waddill and L. Sacks, Another generalized Fibonacci sequence, Fib. Quart., 5 (1967), 209-222.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Eric Weisstein's World of Mathematics, Tribonacci Number

Wikipedia, Generalizations of Fibonacci numbers

Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1,1)

FORMULA

G.f.: x^2/(1 - x - x^2 - x^3).

G.f.: x^2 / (1 - x / (1 - x / (1 + x^2 / (1 + x)))). - Michael Somos, May 12 2012

a(n+1)/a(n) -> A058265.

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, Dec 17 2004

a(n+2) = sum{k=0..n, T(n-k, k)}, T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005

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, May 22 2006

a(n) = 3*c*((1/3)*(a+b+1))^n/(c^2-2*c+4) where a=(19+3*sqrt(33))^(1/3), b=(19-3*sqrt(33))^(1/3), c=(586+102*sqrt(33))^(1/3). The offset is 1. a(3)=2. Round off to the nearest integer. - Al Hakanson (hawkuu(AT)gmail.com), Feb 02 2009

a(n) = 3*((a+b+1)/3)^n/(a^2+b^2+4) where a=(19+3*sqrt(33))^(1/3), b=(19-3*sqrt(33))^(1/3). No offset. a(4)=2. Round off to the nearest integer. - Anton Nikonov

Another form of the g.f.: f(z)=(z^2-z^3)/(1-2*z+z^4). Then we obtain a(n) as sum: a(n)=sum((-1)^i*binomial(n-2-3*i,i)*2^(n-2-4*i),i=0..floor((n-2)/4))-sum((-1)^i*binomial(n-3-3*i,i)*2^(n-3-4*i),i=0..floor((n-3)/4)) with natural convention: sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010

a(n) = sum(k=1..n, sum( i=k..n mod(4*k-i,3)=0, binomial(k,(4*k-i)/3)*(-1)^((i-k)/3)*binomial(n-i+k-1,k-1)). - Vladimir Kruchinin, Aug 18 2010

a(n) = 2*a(n-2)+2*a(n-3)+a(n-4). - Gary Detlefs, Sep 13 2010

sum_{k=0..2*n} A000073(k+b)*A027907(n,k) = A000073(3*n+b), b>=0 (see A099464, A074581).

a(0)=a(1)=0, a(2)=a(3)=1, a(n)=2*a(n-1)-a(n-4). - Vincenzo Librandi, Dec 20 2010

For n>0 Lim_{n -> Inf} a(n-1)/a(n) = 1/((1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3)=0.54368901269... a root of 1-x-x^2-x^3. - Tim Monahan, Aug 13 2011

Starting (1, 2, 4, 7,...) is the INVERT transform of (1, 1, 1, 0, 0, 0,...). - Gary W. Adamson, May 13 2013

G.f.: Q(0)*x^2/2 , where Q(k) = 1 + 1/(1 - x*(4*k+1 + x + x^2)/( x*(4*k+3 + x + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013

EXAMPLE

G.f. = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + 24*x^8 + 44*x^9 + 81*x^10 + ...

MAPLE

A000073:=-z/(-1+z+z**2+z**3); [Simon Plouffe in his 1992 dissertation.]

a:=taylor((z^2-z^3)/(1-2*z+z^4), z=0, 31); for p from 0 to 30 do j(p):=coeff(a, z, p):od :seq(j(p), p=0..30); for n from 0 to 30 do k(n):=sum((-1)^i*binomial(n-2-3*i, i)*2^(n-2-4*i), i=0..floor((n-2)/4))-sum((-1)^i*binomial(n-3-3*i, i)*2^(n-3-4*i), i=0..floor((n-3)/4)):od:seq(k(n), n=0..30); [Richard Choulet, Feb 22 2010]

MATHEMATICA

CoefficientList[Series[x^2/(1 - x - x^2 - x^3), {x, 0, 50}], x]

a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; Array[a, 36, 0] (* Robert G. Wilson v, Nov 07 2010 *)

LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *)

a[ n_] := SeriesCoefficient[ If[ n < 0, x / (1 + x + x^2 - x^3), x^2 / (1 - x - x^2 - x^3)], {x, 0, Abs@n}]; (* Michael Somos, Jun 01 2013 *)

PROG

(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 */

(Maxima) a(n):=sum(sum(if mod(4*k-i, 3)>0 then 0 else binomial(k, (4*k-i)/3)*(-1)^((i-k)/3)*binomial(n-i+k-1, k-1), i, k, n), k, 1, n); \\ Vladimir Kruchinin, Aug 18 2010

(Maxima) A000073[0]:0$

A000073[1]:0$

A000073[2]:1$

A000073[n]:=A000073[n-1]+A000073[n-2]+A000073[n-3]$

  makelist(A000073[n], n, 0, 40);  /* Emanuele Munarini, Mar 01 2011 */

(Haskell)

a000073 n = a000073_list !! n

a000073_list = 0 : 0 : 1 : zipWith (+) a000073_list (tail

                          (zipWith (+) a000073_list $ tail a000073_list))

-- Reinhard Zumkeller, Dec 12 2011

(Python)

def a(n, adict={0:0, 1:0, 2:1}):

.if n in adict:

..return adict[n]

.adict[n]=a(n-1)+a(n-2)+a(n-3)

.return adict[n] # David Nacin, Mar 07 2012

CROSSREFS

Cf. A000045, A000073, A000213, A001590, A081172, A145027, A001644, A063401, A008937, A089068, A027084, A062544, A077902, A054668, A027083, A027024, A118390, A050231, A046738 (Pisano periods).

A057597 is this sequence run backwards: A057597(n) = a(1-n).

Row 3 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

Cf. A000931, A153462, A230216.

Cf. partitions: A240844 and A117546.

See also the tribonacci triangle A008288.

Sequence in context: A107281 A006744 A054175 * A160254 A005318 A102111

Adjacent sequences:  A000070 A000071 A000072 * A000074 A000075 A000076

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000

STATUS

approved

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Last modified November 26 01:37 EST 2014. Contains 250017 sequences.