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

0,1

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).

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)

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 *)

Prog

(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
(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
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (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

Crossrefs

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

Sequence in context: A324224 A264622 A275017 * A294947 A265232 A011016

Adjacent sequences: A141033 A141034 A141035 * A141037 A141038 A141039

Keyword

nonn,easy

Author

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

Extensions

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

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

Status

approved