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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)
279
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 denote the 3 X 3 matrix [0,0,1;1,1,1;0,1,0]. A000073(n) corresponds to both the (1,2) and (3,1) entries 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

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

Least significant bits of A000073 are given in A021913 (a(n) mod 2 = A021913(n)). - Andres Cicuttin, Apr 04 2016

REFERENCES

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.

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 from T. D. Noe)

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

Pornpawee Anantakitpaisal and Kantaphon Kuhapatanakul, Reciprocal Sums of the Tribonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.2.1.

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

J. L. Arocha, B. Llano, The number of dominating k-sets of paths, cycles and wheels, arXiv preprint arXiv:1601.01268 [math.CO], 2016.

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

Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.

D. Birmajer, J. Gil and M. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014.

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv preprint arXiv:1505.06339 [math.NT], 2015.

David Broadhurst, Multiple Landen values and the tribonacci numbers, arXiv:1504.05303 [hep-th], 2015.

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

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

Chekhova, Nataliya, Pascal Hubert, and Ali Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, Journal de théorie des nombres de Bordeaux, 13.2 (2001): 371-394.

Curtis Cooper, S. Miller, P. Moses, M. Sahin, et al., On Identities Of Ruggles, Horadam, Howard, and Young, Preprint, 2016; Proceedings of the 17th International Conference on Fibonacci Numbers and Their Applications, Université de Caen-Normandie, Caen, France, June 26 to July 2, 2016.

Michael Dairyko, Samantha Tyner, Lara Pudwell, Casey Wynn, 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

G. P. B. Dresden, Z. Du, A Simplified Binet Formula for k-Generalized Fibonacci Numbers, J. Int. Seq. 17 (2014) # 14.4.7

M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers & Mathematics with Applications, Vol. 29 (Issue 12, June 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.

Tim Evink, Paul Alexander Helminck, Tribonacci numbers and primes of the form p=x^2+11y^2, arXiv:1801.04605 [math.NT], 2018.

Vinicius Facó, D. Marques, Tribonacci Numbers and the Brocard-Ramanujan Equation, Journal of Integer Sequences, Vol. 19, 2016, #16.4.4.

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

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

Robert Frontczak, Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, Vol. 12 (2018), No. 1, pp. 19-24.

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

Nurettin Irmak, László Szalay, Tribonacci Numbers Close to the Sum 2^a+3^b+5^c, Math. Scand. Vol 118, No 1 (2016).

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, arXiv:physics/0411195 [physics.hist-ph], 2004.

Tamara Kogan, L. Sapir, A. Sapir, A. Sapir, The Fibonacci family of iterative processes for solving nonlinear equations, Applied Numerical Mathematics 110 (2016) 148-158

T. Komatsu, V. Laohakosol, On the Sum of Reciprocals of Numbers Satisfying a Recurrence Relation of Order s, J. Int. Seq. 13 (2010), 10.5.8.

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Pin-Yen Lin, De Moivre type identities for the Tribonacci numbers, The Fibonacci Quarterly 26, no.2, (1988), 131-134

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.

Sepideh Maleki, Martin Burtscher, Automatic Hierarchical Parallelization of Linear Recurrences, Proceedings of the 23rd International Conference on Architectural Support for Programming Languages and Operating Systems, ACM, 2018.

T. Mansour, Permutations avoiding a set of patterns from S_3 and a pattern from S_4, arXiv:math/9909019 [math.CO], 1999.

T. Mansour and M. Shattuck, Polynomials whose coefficients are generalized Tribonacci numbers, Applied Mathematics and Computation, Volume 219, Issue 15, Apr 01 2013, Pages 8366-8374.

T. Mansour, M. Shattuck, A monotonicity property for generalized Fibonacci sequences, arXiv:1410.6943 [math.CO], 2014.

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.

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.

H. Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014) # 14.6.2, odd length middle 0 with r=2.

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.

J. L. Ramírez, V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.

Michel Rigo, Relations on words, arXiv preprint arXiv:1602.03364 [cs.FL], 2016.

A. G. Shannon, Tribonacci numbers and Pascal’s pyramid, part b, Fibonacci Quart. 15(3) (1977) 268-275.

M. Shattuck, Combinatorial identities for incomplete tribonacci polynomials, arXiv preprint arXiv:1406.2755 [math.CO], 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 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

G.f.: Sum_{n >= 0} x^(n+2) *[ Product_{k = 1..n} (k + k*x + x^2)/(1 + k*x + k*x^2) ] = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + ... may be proved by the method of telescoping sums. - Peter Bala, Jan 04 2015

a(n+1)/a(n) -> A058265.  a(n-1)/a(n) -> A192918.

a(n) = central 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... = A058265, 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), where 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

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 (which are 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

a(n+1) = 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). 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). 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 a sum: a(n) = Sum_{i=0..floor((n-2)/4)} ((-1)^i*binomial(n-2-3*i,i)*2^(n-2-4*i)) - Sum_{i=0..floor((n-3)/4)} ((-1)^i*binomial(n-3-3*i,i)*2^(n-3-4*i)) with natural convention: Sum_{i=m..n} alpha(i) = 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} a(k+b)*A027907(n,k) = a(3*n+b), b >= 0 (see A099464, A074581).

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

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

a(n+2) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} binomial(n-2j,k)*binomial(j,k)*2^k. - Tony Foster III, Sep 08 2017

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 := proc(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)) ;

end proc:

seq(A000073(n), n=0..30) ; # Richard Choulet, Feb 22 2010

# second Maple program:

a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3]:

seq(a(n), n=0..40);  # Alois P. Heinz, Dec 19 2016

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 *)

Table[-RootSum[-1 - # - #^2 + #^3 &, -#^n - 9 #^(n + 1) + 4 #^(n + 2) &]/22, {n, 0, 20}] (* Eric W. Weisstein, Nov 09 2017 *)

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 */

(PARI) x='x+O('x^99); concat([0, 0], Vec(x^2/(1-x-x^2-x^3))) \\ Altug Alkan, Apr 04 2016

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3] \\ Charles R Greathouse IV, Apr 18 2016, simplified by M. F. Hasler, Apr 18 2018

(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

(MAGMA) [n le 3 select Floor(n/3) else Self(n-1)+Self(n-2)+Self(n-3): n in [1..70]]; // Vincenzo Librandi, Jan 29 2016

CROSSREFS

Cf. A000045, A000078, A000213, A000931, A001590 (a(n)+a(n+1)), A001644, A008288 (tribonacci triangle), A008937 (partial sums), A021913, A027024, A027083, A027084, A046738 (Pisano periods), A050231, A054668, A062544, A063401, A077902, A081172, A089068, A118390, A145027, A153462, A230216.

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

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

Cf. Partitions: A240844 and A117546.

Cf. A092836 (subsequence of primes), A299399 = A092835 + 1 (indices of primes).

Sequence in context: A107281 A006744 A054175 * A255069 A160254 A276661

Adjacent sequences:  A000070 A000071 A000072 * A000074 A000075 A000076

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Minor edits by M. F. Hasler, Apr 18 2018

STATUS

approved

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Last modified May 24 18:54 EDT 2018. Contains 304533 sequences. (Running on oeis4.)