OFFSET
0,5
COMMENTS
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)}_{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
REFERENCES
Kenneth Edwards, 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).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
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.
W. 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, 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, L. Sapir, A. Sapir, and A. Sapir, The Fibonacci family of iterative processes for solving nonlinear equations, Applied Numerical Mathematics 110 (2016) 148-158.
D. Krob and J.-Y. 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, 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.
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.
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).
FORMULA
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)
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
EXAMPLE
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, 1, 1]^n*[0; 1; 0])[1, 1] \\ Charles R Greathouse IV, Jul 28 2015
(Sage)
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
(Magma) 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
(GAP) 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
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Additional comments from Miklos Kristof, Jul 03 2002
STATUS
approved