A081172 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = 1, a(2) = 0.

1, 1, 0, 2, 3, 5, 10, 18, 33, 61, 112, 206, 379, 697, 1282, 2358, 4337, 7977, 14672, 26986, 49635, 91293, 167914, 308842, 568049, 1044805, 1921696, 3534550, 6501051, 11957297, 21992898, 40451246, 74401441, 136845585, 251698272, 462945298, 851489155

Offset: 0

Author

Harry J. Smith, Apr 19 2003

Keywords

• nonn
• easy

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

Completes the set of tribonacci numbers starting with 0's and 1's in the first three terms:

0,0,0 A000004;

0,0,1 A000073;

0,1,0 A001590;

0,1,1 A000073 starting at a(1);

1,0,0 A000073 starting at a(-1);

1,0,1 A001590;

1,1,0 this sequence;

1,1,1 A000213.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..500
• 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.
• N. G. Voll, Some identities for four term recurrence relations, Fib. Quart., 51 (2013), 268-273.
• Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Crossrefs

Cf. A000004, A000073, A000213, A001590, A020992.

Programs

GAP

a:=[1,1,0];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-2*x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

Maple

A081172 := proc(n)
    option remember;
    if n <= 2 then
        op(n+1,[1,1,0]) ;
    else
        add(procname(n-i),i=1..3) ;
    end if;
end proc: # R. J. Mathar, Aug 09 2012

Mathematica

LinearRecurrence[{1,1,1}, {1,1,0}, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 07 2011 *)

PARI

{ a1=1; a2=1; a3=0; write("b081172.txt",0," ",a1); write("b081172.txt",1," ",a2); write("b081172.txt",2," ",a3); for(n=3,500, a=a1+a2+a3; a1=a2; a2=a3; a3=a; write("b081172.txt",n," ",a) ) } \\ Harry J. Smith, Mar 20 2009

PARI

my(x='x+O('x^40)); Vec((1-2*x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

Sage

((1-2*x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

Formula

From R. J. Mathar, Mar 27 2009: (Start)

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

a(n) = A000073(n+2) - 2*A000073(n). (End)