A106293 -id:A106293 - cp's OEIS frontend

A000213 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, 2209, 4063, 7473, 13745, 25281, 46499, 85525, 157305, 289329, 532159, 978793, 1800281, 3311233, 6090307, 11201821, 20603361, 37895489, 69700671, 128199521, 235795681, 433695873, 797691075, 1467182629

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
Number of (n-1)-bit binary sequences with each one adjacent to a zero. - R. H. Hardin, Dec 24 2007
The binomial transform is A099216. The inverse binomial transform is (-1)^n*A124395(n). - R. J. Mathar, Aug 19 2008
Equals INVERT transform of (1, 0, 2, 0, 2, 0, 2, ...). a(6) = 17 = (1, 1, 1, 3, 5, 9) dot (0, 2, 0, 2, 0, 1) = (0 + 2 + 0 + 6 + 0 + 9) = 17. - Gary W. Adamson, Apr 27 2009
From John M. Campbell, May 16 2011: (Start)
Equals the number of tilings of a 2 X n grid using singletons and "S-shaped tetrominoes" (i.e., shapes of the form Polygon[{{0, 0}, {2, 0}, {2, 1}, {3, 1}, {3, 2}, {1, 2}, {1, 1}, {0, 1}}]).
Also equals the number of tilings of a 2 X n grid using singletons and "T-shaped tetrominoes" (i.e., shapes of the form Polygon[{{0, 0}, {3, 0}, {3, 1}, {2, 1}, {2, 2}, {1, 2}, {1, 1}, {0, 1}}]). (End)
Pisano period lengths: 1, 1, 13, 4, 31, 13, 48, 8, 39, 31, 110, 52, 168, 48, 403, 16, 96, 39, 360, 124, ... (differs from A106293). - R. J. Mathar, Aug 10 2012
a(n) is the number of compositions of n with no consecutive 1's. a(4) = 5 because we have: 4, 3+1, 1+3, 2+2, 1+2+1. Cf. A239791, A003242. - Geoffrey Critzer, Mar 27 2014
a(n+2) is the number of words of length n over alphabet {1,2,3} without having {11,12,22,23} as substrings. - Ran Pan, Sep 16 2015
Satisfies Benford's law [see A186190]. - N. J. A. Sloane, Feb 09 2017
a(n) is also the number of dominating sets on the (n-1)-path graph. - Eric W. Weisstein, Mar 31 2017
a(n) is also the number of maximal irredundant sets and minimal dominating sets in the (2n-3)-triangular snake graph. - Eric W. Weisstein, Jun 09 2019
a(n) is also the number of anti-palindromic compositions of n, where a composition (c(1), c(2),..., c(k)) is anti-palindromic if c(i) is not equal to c(k+1-i) whenever 1 <= i <= k/2. For instance, there are a(4) = 5 anti-palindromic compositions of 4: 4, 31, 13, 211, 112. - Jia Huang, Apr 08 2023

			G.f. = 1 + x + x^2 + 3*x^3 + 5*x^4 + 9*x^5+ 17*x^6 + 31*x^7 + 57*x^8 + ...

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).

Indranil Ghosh, Table of n, a(n) for n = 0..3772 (terms 0..200 from T. D. Noe)
George E. Andrews, Matthew Just, and Greg Simay, Anti-palindromic compositions, arXiv:2102.01613 [math.CO], 2021. Also Fib. Q., 60:2 (2022), 164-176. See Table 1.
Joerg Arndt, Matters Computational (The Fxtbook), p.312
Jean-Luc Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). See Table 4.
Jean-Luc Baril and Nathanaël Hassler, Intervals in a family of Fibonacci lattices, Univ. de Bourgogne (France, 2024). See p. 7.
B. G. Baumgart, Letter to the editor Part 1 Part 2 Part 3, Fib. Quart. 2 (1964), 260, 302.
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.
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, arXiv:1907.06517 [math.CO], 2019.
Mark Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
Nick Hobson, Python program for this sequence
Joanna Jaszunska and Jan Okninski, Structure of Chinese algebras, Journal of Algebra, Volume 346, Issue 1, 15 November 2011, Pages 31-81.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
I. Tasoulas, K. Manes, A. Sapounakis, and P. Tsikouras, Chains with Small Intervals in the Lattice of Binary Paths, arXiv:1911.10883 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Irredundant Set
Eric Weisstein's World of Mathematics, Minimal Dominating Set
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Triangular Snake Graph
Eric Weisstein's World of Mathematics, Tribonacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1).
Index entries for sequences related to Benford's law

Cf. A000073, A000288, A000322, A000383, A046735, A060455, A180735, A186190.

a:=[1,1,1];; for n in [4..45] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 09 2019

a000213 n = a000213_list !! n
a000213_list = 1 : 1 : 1 : zipWith (+) a000213_list
   (tail $ zipWith (+) a000213_list (tail a000213_list))
-- Reinhard Zumkeller, Apr 07 2012

I:=[1,1,1]; [n le 3 select I[n] else Self(n-1) + Self(n-2) + Self(n-3): n in [1..45]]; // G. C. Greubel, Jun 09 2019

K:=(1-z^2)/(1-z-z^2-z^3): Kser:=series(K, z=0, 45): seq((coeff(Kser, z, n)), n= 0..34); # Zerinvary Lajos, Nov 08 2007
A000213:=(z-1)*(1+z)/(-1+z+z**2+z**3); # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1, 1, 1}, {1, 1, 1}, 45] (* Harvey P. Dale, May 23 2011 *)
Table[RootSum[-1 - # - #^2 + #^3 &, 2 #^n - 4 #^(n + 1) + 3 #^(n + 2) &]/11, {n, 0, 45}] (* Eric W. Weisstein, Apr 10 2018 *)
CoefficientList[Series[(1-x)(1+x)/(1-x-x^2-x^3), {x, 0, 45}], x] (* Eric W. Weisstein, Apr 10 2018 *)

a(n):=sum(sum(binomial(n-2*m+1,m-i)*binomial(n-2*m+i,n-2*m), i,0,m),m,0,(n)/2); /* Vladimir Kruchinin, Dec 17 2011 */

a(n)=tn=[1,1,1;1,0,0;0,1,0]^n;tn[3,1]+tn[3,2]+tn[3,3] \\ Charles R Greathouse IV, Feb 18 2011

alst = [1, 1, 1]
[alst.append(alst[n-1] + alst[n-2] + alst[n-3]) for n in range(3, 37)]
print(alst) # Michael S. Branicky, Sep 21 2021

((1-x^2)/(1-x-x^2-x^3)).series(x, 45).coefficients(x, sparse=False) # G. C. Greubel, Jun 09 2019

G.f.: (1-x)*(1+x)/(1-x-x^2-x^3). - Ralf Stephan, Feb 11 2004
G.f.: 1 / (1 - x / (1 - 2*x^2 / (1 + x^2))). - Michael Somos, May 12 2012
a(n) = rightmost term of M^n * [1 1 1], where M is the 3 X 3 matrix [1 1 1 / 1 0 0 / 0 1 0]. M^n * [1 1 1] = [a(n+2) a(n+1) a(n)]. 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) = A001590(n+3) - A001590(n+2); a(n+1) - a(n) = 2*A000073(n); a(n) = A000073(n+3) - A000073(n+1). - Reinhard Zumkeller, May 22 2006
a(n) = A001590(n) + A001590(n+1). - Philippe Deléham, Sep 25 2006
a(n) ~ (F - 1) * T^n, where F = A086254 and T = A058265. - Charles R Greathouse IV, Nov 09 2008
a(n) = 2*a(n-1) - a(n-4), n > 3. - Gary Detlefs, Sep 13 2010
a(n) = Sum_{m=0..n/2} Sum_{i=0..m} binomial(n-2*m+1,m-i)*binomial(n-2*m+i, n-2*m). - Vladimir Kruchinin, Dec 17 2011
a(n) = 2*A008937(n-2) + 1 for n > 1. - Reinhard Zumkeller, Apr 07 2012
G.f.: 1+x/(U(0) - x) where U(k) = 1 - x^2/(1 - 1/(1 + 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 16 2012
G.f.: 1 + x + x^2/(G(0)-x) where G(k) = 1 - x*(2*k+1)/(1 - 1/(1 + (2*k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 17 2012
G.f.: (1+x)*(1-x)*(1 + x*(G(0)-1)/(x+1)) where G(k) = 1 + (1+x+x^2)/(1-x/(x+1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
G.f.: 1/(1+x-G(0)), where G(k) = 1 - 1/(1 - x/(x - 1/(1 + 1/(1 - x/(x + 1/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-1)^n * A180735(-1-n) for all n in Z. - Michael Somos, Aug 15 2015
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-r^2+2*r+2). - Fabian Pereyra, Nov 21 2024

A001644 a(n) = a(n-1) + a(n-2) + a(n-3), a(0)=3, a(1)=1, a(2)=3.

Original entry on oeis.org

3, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499, 2757, 5071, 9327, 17155, 31553, 58035, 106743, 196331, 361109, 664183, 1221623, 2246915, 4132721, 7601259, 13980895, 25714875, 47297029, 86992799, 160004703, 294294531, 541292033, 995591267, 1831177831

Offset: 0

N. J. A. Sloane

For n >= 3, a(n) is the number of cyclic sequences consisting of n zeros and ones that do not contain three consecutive ones provided the positions of the zeros and ones are fixed on a circle. This is proved in Charalambides (1991) and Zhang and Hadjicostas (2015). For example, a(3)=7 because only the sequences 110, 101, 011, 001, 010, 100 and 000 avoid three consecutive ones. (For n=1,2 the statement is still true provided we allow the sequence to wrap around itself on a circle.) - Petros Hadjicostas, Dec 16 2016
For n >= 3, also the number of dominating sets on the n-cycle graph C_n. - Eric W. Weisstein, Mar 30 2017
For n >= 3, also the number of minimal dominating sets and maximal irredundant sets on the n-sun graph. - Eric W. Weisstein, Jul 28 and Aug 17 2017
For n >= 3, also the number of minimal edge covers in the n-web graph. - Eric W. Weisstein, Aug 03 2017
For n >= 1, also the number of ways to tile a bracelet of length n with squares, dominoes, and trominoes. - Ruijia Li and Greg Dresden, Sep 14 2019
If n is prime, then a(n)-1 is a multiple of n ; a counterexample for the converse is given by n = 182. - Robert FERREOL, Apr 03 2024

			G.f. = 3 + x + 3*x^2 + 7*x^3 + 11*x^4 + 21*x^5 + 39*x^6 + 71*x^7 + 131*x^8 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 500.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
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).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 1..200 from T. D. Noe)
Kunle Adegoke, Robert Frontczak, and Taras Goy, Binomial Tribonacci Sums, Disc. Math. Lett. (2022) Vol. 8, 30-37.
Kunle Adegoke, Adenike Olatinwo, and Winning Oyekanmi, New Tribonacci Recurrence Relations and Addition Formulas, arXiv:1811.03663 [math.CO], 2018.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv:1505.06339 [math.NT], 2015.
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.
C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
Curtis Cooper, Steven Miller, Peter J. C. Moses, Murat Sahin, and Thotsaporn Thanatipanonda, On Identities Of Ruggles, Horadam, Howard, and Young, Preprint, 2016.
Curtis Cooper, Steven Miller, Peter J. C. Moses, Murat Sahin, and Thotsaporn Thanatipanonda, On identities of Ruggles, Hradam, Howard, and Young, The Fibonacci Quarterly, 55.5 (2017), 42-65.
M. Elia, Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly, 39.2 (2001), 107-109.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3.
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Robert Frontczak, Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, Vol. 12 (2018), No. 1, pp. 19-24.
Robert Frontczak, Convolutions for Generalized Tribonacci Numbers and Related Results, International Journal of Mathematical Analysis (2018) Vol. 12, Issue 7, 307-324.
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), #16.8.2.
A. Ilic, S. Klavzar, and Y. Rho, Generalized Lucas Cubes, Appl. An. Disc. Math. 6 (2012) 82-94, proposition 11 for the sequence starting 1, 2, 4, 7, 11,...
Günter Köhler, Generating functions of Fibonacci-like sequences and decimal expansion of some fractions, The Fibonacci Quarterly 23, no.1, (1985), 29-35 [a(n) is there Lambda_n on p. 35].
Pin-Yen Lin, De Moivre type identities for the Tribonacci numbers, The Fibonacci Quarterly 26, no.2, (1988), 131-134.
Matthew Macauley, Jon McCammond, and Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
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.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 21.
J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011) # 11.3.4, remark 17.
Andreas N. Philippou and Spiros D. Dafnis, A simple proof of an identity generalizing Fibonacci-Lucas identities, Fibonacci Quarterly (2018) Vol. 56, No. 4, 334-336.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. L. Ramírez and V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.
Souvik Roy, Nazim Fatès, and Sukanta Das, Reversibility of Elementary Cellular Automata with fully asynchronous updating: an analysis of the rules with partial recurrence, hal-04456320 [nlin.CG], [cs], 2024. See p. 17.
S. Saito, T. Tanaka, and N. Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 3.
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Lucas n-Step Number
Eric Weisstein's World of Mathematics, Maximal Irredundant Set
Eric Weisstein's World of Mathematics, Minimal Edge Cover
Eric Weisstein's World of Mathematics, Sun Graph
Eric Weisstein's World of Mathematics, Tribonacci Number
Eric Weisstein's World of Mathematics, Web Graph
Wikipedia, Companion matrix.
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A073145, A106293 (Pisano periods), A073728 (partial sums).
Cf. A058265.
Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649.

a:=[3,1,3];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Dec 18 2018

a001644 n = a001644_list !! n
a001644_list = 3 : 1 : 3 : zipWith3 (((+) .) . (+))
               a001644_list (tail a001644_list) (drop 2 a001644_list)
-- Reinhard Zumkeller, Apr 13 2014

I:=[3,1,3]; [n le 3 select I[n] else Self(n-1)+Self(n-2)+ Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 04 2017

A001644:=-(1+2*z+3*z**2)/(z**3+z**2+z-1); # Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 3
A001644 :=proc(n)
    option remember;
    if n <= 2 then
        1+2*modp(n+1,2)
    else
        procname(n-1)+procname(n-2)+procname(n-3);
    end if;
end proc:
seq(A001644(n),n=0..80) ;

a[x_]:= a[x] = a[x-1] +a[x-2] +a[x-3]; a[0] = 3; a[1] = 1; a[2] = 3; Array[a, 40, 0]
a[n_]:= n*Sum[Sum[Binomial[j, n-3*k+2*j]*Binomial[k, j], {j,n-3*k,k}]/k, {k, n}]; a[0] = 3; Array[a, 40, 0] (* Robert G. Wilson v, Feb 24 2011 *)
LinearRecurrence[{1, 1, 1}, {3, 1, 3}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
Table[RootSum[-1 - # - #^2 + #^3 &, #^n &], {n, 0, 40}] (* Eric W. Weisstein, Mar 30 2017 *)
RootSum[-1 - # - #^2 + #^3 &, #^Range[0, 40] &] (* Eric W. Weisstein, Aug 17 2017 *)

{a(n) = if( n<0, polsym(1 - x - x^2 - x^3, -n)[-n+1], polsym(1 + x + x^2 - x^3, n)[n+1])}; /* Michael Somos, Nov 02 2002 */

my(x='x+O('x^40)); Vec((3-2*x-x^2)/(1-x-x^2-x^3)) \\ Altug Alkan, Apr 19 2018

((3-2*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Mar 22 2019

Binet's formula: a(n) = r1^n + r2^n + r3^n, where r1, r2, r3 are the roots of the characteristic polynomial 1 + x + x^2 - x^3, see A058265.
a(n) = A000073(n) + 2*A000073(n-1) + 3*A000073(n-2).
G.f.: (3-2*x-x^2)/(1-x-x^2-x^3). - Miklos Kristof, Jul 29 2002
a(n) = n*Sum_{k=1..n} Sum_{j=n-3*k..k} binomial(j, n-3*k+2*j)*binomial(k,j)/k, n > 0, a(0)=3. - Vladimir Kruchinin, Feb 24 2011
a(n) = a(n-1) + a(n-2) + a(n-3), a(0)=3, a(1)=1, a(2)=3. - Harvey P. Dale, Feb 01 2015
a(n) = A073145(-n). for all n in Z. - Michael Somos, Dec 17 2016
Sum_{k=0..n} k*a(k) = (n*a(n+3) - a(n+2) - (n+1)*a(n+1) + 4)/2. - Yichen Wang, Aug 30 2020
a(n) = Trace(M^n), where M = [0, 0, 1; 1, 0, 1; 0, 1, 1] is the companion matrix to the monic polynomial x^3 - x^2 - x - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Dec 29 2022

Edited by Mario Catalani (mario.catalani(AT)unito.it), Jul 17 2002

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step recursions, A000073 and A001644. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -44 = -4*11. The primes p yielding 3 distinct zeros, A106279, correspond to the periods of the sequences A000073(k) mod p and A001644(k) mod p having length less than p. The Lucas 3-step sequence mod p has two additional primes p for which the period is less than p: 2 and 11, which are factors of the discriminant -44. For p=11, the Fibonacci 3-step sequence mod p has a period of p(p-1).

This sequence differs from the corresponding Fibonacci sequence (A106302) at n=1 and 5 because these correspond to the primes 2 and 11, which are the prime factors of -44, the discriminant of the characteristic polynomial x^3-x^2-x-1. We have a(n) < prime(n) for the primes 2, 11 and A106279.

For a prime p, the period depends on the zeros of x^3-x^2-x-1 mod p. If there are 3 zeros, then the period is < p. If there are no zeros, then the period is p^2+p+1 or a simple fraction of p^2+p+1. Also note that the period can be prime, as for p=3, 5, 31, 59, 71, 89, 97, 157, 223. When the period is prime, the orbits have a simple structure. [From T. D. Noe, Sep 18 2008]

cp's OEIS Frontend

A000213 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=a(2)=1.

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A001644 a(n) = a(n-1) + a(n-2) + a(n-3), a(0)=3, a(1)=1, a(2)=3.

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A106276 Number of distinct zeros of x^3-x^2-x-1 mod prime(n).

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A106294 Period of the Lucas 3-step sequence A001644 mod prime(n).

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