A058255 -id:A058255 - cp's OEIS frontend

A001590 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=0, a(1)=1, a(2)=0.

0, 1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, 423, 778, 1431, 2632, 4841, 8904, 16377, 30122, 55403, 101902, 187427, 344732, 634061, 1166220, 2145013, 3945294, 7256527, 13346834, 24548655, 45152016, 83047505, 152748176, 280947697, 516743378, 950439251

Offset: 0

N. J. A. Sloane

The name "tribonacci number" is less well-defined than "Fibonacci number". The sequence A000073 (which begins 0, 0, 1) is probably the most important version, but the name has also been applied to A000213, A001590, and A081172. - N. J. A. Sloane, Jul 25 2024
Dimensions of the homogeneous components of the higher order peak algebra associated to cubic roots of unity (Hilbert series = 1 + 1*t + 2*t^2 + 3*t^3 + 6*t^4 + 11*t^5 ...). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 22 2006
Starting with offset 3: (1, 2, 3, 6, 11, 10, 37, ...) = row sums of triangle A145579. - Gary W. Adamson, Oct 13 2008
Starting (1, 2, 3, 6, 11, ...) = INVERT transform of the periodic sequence (1, 1, 0, 1, 1, 0, 1, 1, 0, ...). - Gary W. Adamson, May 04 2009
The comment of May 04 2009 is equivalent to: The numbers of ordered compositions of n using integers that are not multiples of 3 is equal to a(n+2) for n>=1, see [Hoggatt-Bicknell (1975) eq (2.7)]. - Gary W. Adamson, May 13 2013
Primes in the sequence are 2, 3, 11, 37, 634061, 7256527, ... in A231574. - R. J. Mathar, Aug 09 2012
Pisano period lengths: 1, 2, 13, 8, 31, 26, 48, 16, 39, 62,110,104,168, 48,403, 32, 96, 78, 360, 248, ... . - R. J. Mathar, Aug 10 2012
a(n+1) is the top left entry of the n-th power of any of 3 X 3 matrices [0, 1, 0; 1, 1, 1; 1, 0, 0], [0, 1, 1; 1, 1, 0; 0, 1, 0], [0, 1, 1; 0, 0, 1; 1, 0, 1] or [0, 0, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n+3) equals the number of n-length binary words avoiding runs of zeros of lengths 3i+2, (i=0,1,2,...). - Milan Janjic, Feb 26 2015
Sums of pairs of successive terms of A000073. - N. J. A. Sloane, Oct 30 2016
The power Q^n, for n >= 0, of the tribonacci Q-matrix Q = matrix([1, 1, 1], [1, 0, 0], [0, 1, 0]) is, by the Cayley-Hamilton theorem, Q^n = matrix([a(n+2), a(n+1) + a(n), a(n+1)], [a(n+1), a(n) + a(n-1), a(n)], [a(n), a(n-1) + a(n-2), a(n-1)]), with a(-2) = -1 and a(-1) = 1. One can use a(n) = a(n-1) + a(n-2) + a(n-3) in order to obtain a(-1) and a(-2). - Wolfdieter Lang, Aug 13 2018
a(n+2) is the number of entries n, for n>=1, in the sequence {A278038(k)}A278038(0)%20=%201).%20-%20_Wolfdieter%20Lang">{k>=1} (without A278038(0) = 1). - _Wolfdieter Lang, Sep 11 2018
In terms of the tribonacci numbers T(n) = A000073(n) the nonnegative powers of the Q-matrix (from the Aug 13 2018 comment) are Q^n = T(n)*Q^2 + (T(n-1) + T(n-2))*Q + T(n-1)*1_3, for n >= 0, with T(-1) = 1, T(-2) = -1. This is equivalent to the powers t^n of the tribonacci constant t = A058255 (or also for powers of the complex solutions). - Wolfdieter Lang, Oct 24 2018

			a(12)=a(11)+a(10)+a(9): 230=125+68+37.
For n=5 the partitions of 5 are 1+1+1+1+1 (1 composition), 1+1+1+2 (4 compositions), 1+2+2 (3 compositions), 1+1+3 (not contrib because 3 is a part), 2+3 (no contrib because 3 is a part), 1+4 (2 compositions) and 5 (1 composition), total 1+4+3+2+1=11 =a(5+2) - _R. J. Mathar_, Jan 13 2023

Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
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).

T. D. Noe, Table of n, a(n) for n = 0..200
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See p. 8.
Barry Balof, Restricted tilings and bijections, J. Integer Seq. 15 (2012), no. 2, Article 12.2.3, 17 pp.
Matthias Beck and Neville Robbins, Variations on a Generatingfunctional Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT], 2014.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.
M. Feinberg, New slants, Fib. Quart. 2 (1964), 223-227.
Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fib. Quart 13 (4) (1975) 357, eq (2.7)
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 401
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Tamara Kogan, Luba Sapir, Amir Sapir, and Ariel Sapir, The Fibonacci family of iterative processes for solving nonlinear equations, Applied Numerical Mathematics 110 (2016) 148-158.
Daniel Krob and Jean-Yves Thibon, Higher order peak algebras, arXiv:math/0411407 [math.CO], 2004.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
Sepideh Maleki and 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.
M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7. P_3(n) = 1,2,2,3,3,6,6,11,11,...
Helmut Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014) # 14.6.2, even length, r=2.
Karim Ritter von Merkl, Computing colored Khovanov homology, arXiv:2505.03916 [math.QA], 2025. See p. 13.
Neville Robbins, On Tribonacci Numbers and 3-Regular Compositions, Fibonacci Quart. 52 (2014), no. 1, 16-19. See Adamson's comments.
Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719.
M. E. Waddill and L. Sacks, Another generalized Fibonacci sequence, Fib. Quart., 5 (1967), 209-222.
Eric Weisstein's World of Mathematics, Tribonacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000045, A000073, A027907, A027053, A078042, A145579, A278038.

a:=[0,1,0];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Oct 24 2018

I:=[0,1,0]; [n le 3 select I[n]  else Self(n-1)+Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 19 2018

seq(coeff(series(x*(1-x)/(1-x-x^2-x^3),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 24 2018
# alternative
A001590 := proc(n)
    option remember;
    if n <=2 then
        op(n+1,[0,1,0]) ;
    else
        procname(n-1)+procname(n-2)+procname(n-3) ;
    end if;
end proc:
seq(A001590(n),n=0..30) ;# R. J. Mathar, Nov 22 2024

LinearRecurrence[{1,1,1}, {0,1,0}, 50] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[2]==0, a[n]==a[n-1]+a[n-2]+a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Apr 19 2018 *)

a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[0;1;0])[1,1] \\ Charles R Greathouse IV, Jul 28 2015

def A001590():
    W = [0, 1, 0]
    while True:
        yield W[0]
        W.append(sum(W))
        W.pop(0)
a = A001590(); [next(a) for  in range(38)]  # _Peter Luschny, Sep 12 2016

G.f.: x*(1-x)/(1-x-x^2-x^3).
Limit a(n)/a(n-1) = t where t is the real solution of t^3 = 1 + t + t^2, t = A058265 = 1.839286755... . If T(n) = A000073(n) then t^n  = T(n-1) + a(n)*t + T(n)*t^2, for n >= 0, with T(-1) = 1.
a(3*n) = Sum_{k+l+m=n} (n!/k!l!m!)*a(l+2*m). Example: a(12)=a(8)+4a(7)+10a(6)+16a(5)+19a(4)+16a(3)+10a(2)+4a(1)+a(0) The coefficients are the trinomial coefficients. T(n) and T(n-1) also satisfy this equation. (T(-1)=1)
From Reinhard Zumkeller, May 22 2006: (Start)
a(n) = A000073(n+1)-A000073(n);
a(n) = A000073(n-1)+A000073(n-2) for n>1;
A000213(n-2) = a(n+1)-a(n) for n>1. (End)
a(n) + a(n+1) = A000213(n). - Philippe Deléham, Sep 25 2006
If p[1]=0, p[i]=2, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1)=det A. - Milan Janjic, May 02 2010
For n>=4, a(n)=2*a(n-1)-a(n-4). - Bob Selcoe, Feb 18 2014
a(-1-n) = -A078046(n). - Michael Somos, Jun 01 2014
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(3*r+1). - Fabian Pereyra, Nov 22 2024

Additional comments from Miklos Kristof, Jul 03 2002

A058254 a(n) = lcm{prime(i)-1, i=1..n}.

Original entry on oeis.org

1, 1, 2, 4, 12, 60, 60, 240, 720, 7920, 55440, 55440, 55440, 55440, 55440, 1275120, 16576560, 480720240, 480720240, 480720240, 480720240, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400, 532748591575200

Offset: 0

Labos Elemer, Dec 06 2000

nonn

A002110(n) divides b^(a(n)+1) - b for every integer b. - Thomas Ordowski, Nov 24 2014
What is the asymptotic growth of this sequence? a(n) <= A005867(n) <= A002110(n) < e^((1 + o(1))n log n) but this is a large overestimate. - Charles R Greathouse IV, Dec 03 2014
Alexander Kalmynin gives a proof that log a(n) = O(p log log p/log p) where p is the n-th prime, see the MathOverflow link. - Charles R Greathouse IV, Sep 17 2021

			For n = 5 and 6: a(5) = a(6) = LCM[1, 2, 4, 6, 10, 12] = 60.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
MathOverflow, Asymptotics of lcm((2-1), (3-1), (5-1), (7-1), (11-1), ..., pn-1)

Cf. A000010, A000142, A002110, A003418, A005867, A006093, A055769, A058255.

a058254 n = a058254_list !! (n-1)
a058254_list = scanl1 lcm a006093_list
-- Reinhard Zumkeller, May 01 2013

seq(ilcm(seq(ithprime(i)-1,i=1..n)), n=0..100); # Robert Israel, Nov 24 2014

Table[LCM @@ (Prime@ Range[1, n] - 1), {n, 27}] (* Michael De Vlieger, Dec 31 2016 *)

a(n)=lcm(apply(p->p-1, primes(n))) \\ Charles R Greathouse IV, Dec 03 2014

a(n) = A002322(A002110(n)). - Thomas Ordowski, Nov 24 2014

Offset corrected by Reinhard Zumkeller, May 01 2013

a(0)=1 prepended by Alois P. Heinz, Apr 01 2021

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A001590 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=0, a(1)=1, a(2)=0.

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A058254 a(n) = lcm{prime(i)-1, i=1..n}.

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A095365 Primes p such that f(q) < f(p) for all primes q < p, where f(p) = lcm(2-1, 3-1, 5-1, 7-1, 11-1, ..., p-1).

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