A141036 Tribonacci-like sequence; a(0)=2, a(1)=1, a(2)=1, a(n) = a(n-1) + a(n-2) + a(n-3).

2, 1, 1, 4, 6, 11, 21, 38, 70, 129, 237, 436, 802, 1475, 2713, 4990, 9178, 16881, 31049, 57108, 105038, 193195, 355341, 653574, 1202110, 2211025, 4066709, 7479844, 13757578, 25304131, 46541553, 85603262, 157448946, 289593761

Offset: 0

Author

Matt Wynne (matwyn(AT)verizon.net), Jul 30 2008

Keywords

• nonn
• easy

Comments

I used the short MATLAB program from the zip file link altered to produce a Lucas version of the tribonacci numbers.

No term is divisible by 8 or 9. - Vladimir Joseph Stephan Orlovsky, Mar 24 2011

a(A246517(n)) = A246518(n). - Reinhard Zumkeller, Sep 15 2014

References

• Martin Gardner, Mathematical Circus, Random House, New York, 1981, p. 165.

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• 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.
• T.-X. He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2
• Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Crossrefs

Cf. A000073, A000213, A001644 (Lucas tribonacci sequence), A246517, A246518.

Programs

Haskell

a141036 n = a141036_list !! n
a141036_list = 2 : 1 : 1 : zipWith3 (((+) .) . (+))
   a141036_list (tail a141036_list) (drop 2 a141036_list)
-- Reinhard Zumkeller, Sep 15 2014

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-x-2*x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 22 2019

Mathematica

a[0]=2; a[1]=1; a[2]=1; a[n_]:= a[n]=a[n-1]+a[n-2]+a[n-3]; Table[a[n], {n, 0, 40}] (* Alonso del Arte, Mar 24 2011 *)
LinearRecurrence[{1, 1, 1}, {2, 1, 1}, 40] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

PARI

a(n)=([0,1,0;0,0,1;1,1,1]^n*[2;1;1])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

PARI

my(x='x+O('x^40)); Vec((2-x-2*x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 22 2019

Sage

((2-x-2*x^2)/(1-x-x^2-x^3)).series(x, 41).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

Formula

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

From R. J. Mathar, Aug 04 2008: (Start)

a(n) = 2*A000213(n) - A000073(n+1).

O.g.f.: (2-x-2*x^2)/(1-x-x^2-x^3). (End)

Extensions

Corrected offset and indices in formulas, R. J. Mathar, Aug 05 2008

Better name from T. D. Noe, Aug 06 2008